# Zero-Based Catalan Recurrence, Exact Divisibility and Both Closed Forms

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261111-r05-catalan-numbers-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Catalan Generation Starts with C Zero Equal to One

Use natural index n=0≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(0, [1], 1, C_zero_equals_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (0, [1], 1, C_zero_equals_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0001/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 2. Catalan Generation Starts with C Zero Equal to One

Use natural index n=1≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(1, [1, 1], 1, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (1, [1, 1], 1, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0002/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 3. Catalan Generation Starts with C Zero Equal to One

Use natural index n=2≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(2, [1, 1, 2], 2, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (2, [1, 1, 2], 2, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0003/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 4. Catalan Generation Starts with C Zero Equal to One

Use natural index n=3≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0004/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 5. Catalan Generation Starts with C Zero Equal to One

Use natural index n=4≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(4, [1, 1, 2, 5, 14], 14, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (4, [1, 1, 2, 5, 14], 14, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0005/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 6. Catalan Generation Starts with C Zero Equal to One

Use natural index n=5≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(5, [1, 1, 2, 5, 14, 42], 42, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (5, [1, 1, 2, 5, 14, 42], 42, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0006/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 7. Catalan Generation Starts with C Zero Equal to One

Use natural index n=6≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(6, [1, 1, 2, 5, 14, 42, 132], 132, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (6, [1, 1, 2, 5, 14, 42, 132], 132, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0007/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 8. Catalan Generation Starts with C Zero Equal to One

Use natural index n=7≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(7, [1, 1, 2, 5, 14, 42, 132, 429], 429, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (7, [1, 1, 2, 5, 14, 42, 132, 429], 429, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0008/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 9. Catalan Generation Starts with C Zero Equal to One

Use natural index n=8≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(8, [1, 1, 2, 5, 14, 42, 132, 429, 1430], 1430, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (8, [1, 1, 2, 5, 14, 42, 132, 429, 1430], 1430, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0009/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 10. Catalan Generation Starts with C Zero Equal to One

Use natural index n=9≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(9, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862], 4862, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (9, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862], 4862, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0010/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 11. Catalan Generation Starts with C Zero Equal to One

Use natural index n=10≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(10, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796], 16796, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (10, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796], 16796, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0011/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 12. Catalan Generation Starts with C Zero Equal to One

Use natural index n=11≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(11, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786], 58786, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (11, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786], 58786, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0012/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 13. Catalan Generation Starts with C Zero Equal to One

Use natural index n=12≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(12, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012], 208012, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (12, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012], 208012, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0013/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 14. Catalan Generation Starts with C Zero Equal to One

Use natural index n=13≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(13, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900], 742900, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (13, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900], 742900, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0014/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 15. Catalan Generation Starts with C Zero Equal to One

Use natural index n=14≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(14, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440], 2674440, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (14, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440], 2674440, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0015/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 16. Catalan Generation Starts with C Zero Equal to One

Use natural index n=15≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(15, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845], 9694845, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (15, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845], 9694845, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0016/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 17. Catalan Generation Starts with C Zero Equal to One

Use natural index n=16≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(16, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670], 35357670, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (16, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670], 35357670, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0017/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 18. Catalan Generation Starts with C Zero Equal to One

Use natural index n=17≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(17, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790], 129644790, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (17, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790], 129644790, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0018/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 19. Catalan Generation Starts with C Zero Equal to One

Use natural index n=18≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(18, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700], 477638700, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (18, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700], 477638700, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0019/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 20. Catalan Generation Starts with C Zero Equal to One

Use natural index n=19≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(19, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190], 1767263190, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (19, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190], 1767263190, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0020/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 21. Catalan Generation Starts with C Zero Equal to One

Use natural index n=20≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(20, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420], 6564120420, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (20, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420], 6564120420, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0021/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 22. Catalan Generation Starts with C Zero Equal to One

Use natural index n=21≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(21, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020], 24466267020, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (21, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020], 24466267020, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0022/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 23. Catalan Generation Starts with C Zero Equal to One

Use natural index n=22≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(22, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640], 91482563640, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (22, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640], 91482563640, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0023/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 24. Catalan Generation Starts with C Zero Equal to One

Use natural index n=23≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(23, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650], 343059613650, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (23, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650], 343059613650, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0024/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 25. Catalan Generation Starts with C Zero Equal to One

Use natural index n=24≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, prefix, C_n, index_convention).

View answer and explanation(24, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324], 1289904147324, C_next_sum_i_zero_to_n_minus_one)

Generate the complete prefix through n by summing every split in the convolution recurrence. Include the base case and retain exact integer strings, not floating approximations. The exact result, retaining the requested order, is (24, [1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324], 1289904147324, C_next_sum_i_zero_to_n_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0025/)

Catalan numbers · qf-mod42-catalan-recurrence-generation · candidate





## 26. Check Divisibility before Taking the Catalan Quotient

Use natural index n=0≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(0, 1, 1, 0, 1, 1, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (0, 1, 1, 0, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0026/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 27. Check Divisibility before Taking the Catalan Quotient

Use natural index n=1≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(1, 2, 2, 0, 1, 1, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (1, 2, 2, 0, 1, 1, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0027/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 28. Check Divisibility before Taking the Catalan Quotient

Use natural index n=2≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(2, 6, 3, 0, 2, 2, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (2, 6, 3, 0, 2, 2, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0028/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 29. Check Divisibility before Taking the Catalan Quotient

Use natural index n=3≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(3, 20, 4, 0, 5, 5, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (3, 20, 4, 0, 5, 5, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0029/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 30. Check Divisibility before Taking the Catalan Quotient

Use natural index n=4≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(4, 70, 5, 0, 14, 14, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (4, 70, 5, 0, 14, 14, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0030/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 31. Check Divisibility before Taking the Catalan Quotient

Use natural index n=5≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(5, 252, 6, 0, 42, 42, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (5, 252, 6, 0, 42, 42, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0031/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 32. Check Divisibility before Taking the Catalan Quotient

Use natural index n=6≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(6, 924, 7, 0, 132, 132, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (6, 924, 7, 0, 132, 132, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0032/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 33. Check Divisibility before Taking the Catalan Quotient

Use natural index n=7≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(7, 3432, 8, 0, 429, 429, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (7, 3432, 8, 0, 429, 429, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0033/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 34. Check Divisibility before Taking the Catalan Quotient

Use natural index n=8≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(8, 12870, 9, 0, 1430, 1430, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (8, 12870, 9, 0, 1430, 1430, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0034/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 35. Check Divisibility before Taking the Catalan Quotient

Use natural index n=9≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(9, 48620, 10, 0, 4862, 4862, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (9, 48620, 10, 0, 4862, 4862, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0035/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 36. Check Divisibility before Taking the Catalan Quotient

Use natural index n=10≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(10, 184756, 11, 0, 16796, 16796, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (10, 184756, 11, 0, 16796, 16796, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0036/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 37. Check Divisibility before Taking the Catalan Quotient

Use natural index n=11≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(11, 705432, 12, 0, 58786, 58786, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (11, 705432, 12, 0, 58786, 58786, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0037/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 38. Check Divisibility before Taking the Catalan Quotient

Use natural index n=12≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(12, 2704156, 13, 0, 208012, 208012, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (12, 2704156, 13, 0, 208012, 208012, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0038/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 39. Check Divisibility before Taking the Catalan Quotient

Use natural index n=13≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(13, 10400600, 14, 0, 742900, 742900, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (13, 10400600, 14, 0, 742900, 742900, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0039/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 40. Check Divisibility before Taking the Catalan Quotient

Use natural index n=14≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(14, 40116600, 15, 0, 2674440, 2674440, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (14, 40116600, 15, 0, 2674440, 2674440, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0040/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 41. Check Divisibility before Taking the Catalan Quotient

Use natural index n=15≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(15, 155117520, 16, 0, 9694845, 9694845, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (15, 155117520, 16, 0, 9694845, 9694845, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0041/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 42. Check Divisibility before Taking the Catalan Quotient

Use natural index n=16≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(16, 601080390, 17, 0, 35357670, 35357670, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (16, 601080390, 17, 0, 35357670, 35357670, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0042/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 43. Check Divisibility before Taking the Catalan Quotient

Use natural index n=17≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(17, 2333606220, 18, 0, 129644790, 129644790, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (17, 2333606220, 18, 0, 129644790, 129644790, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0043/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 44. Check Divisibility before Taking the Catalan Quotient

Use natural index n=18≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(18, 9075135300, 19, 0, 477638700, 477638700, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (18, 9075135300, 19, 0, 477638700, 477638700, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0044/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 45. Check Divisibility before Taking the Catalan Quotient

Use natural index n=19≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(19, 35345263800, 20, 0, 1767263190, 1767263190, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (19, 35345263800, 20, 0, 1767263190, 1767263190, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0045/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 46. Check Divisibility before Taking the Catalan Quotient

Use natural index n=20≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(20, 137846528820, 21, 0, 6564120420, 6564120420, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (20, 137846528820, 21, 0, 6564120420, 6564120420, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0046/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 47. Check Divisibility before Taking the Catalan Quotient

Use natural index n=21≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(21, 538257874440, 22, 0, 24466267020, 24466267020, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (21, 538257874440, 22, 0, 24466267020, 24466267020, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0047/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 48. Check Divisibility before Taking the Catalan Quotient

Use natural index n=22≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(22, 2104098963720, 23, 0, 91482563640, 91482563640, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (22, 2104098963720, 23, 0, 91482563640, 91482563640, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0048/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 49. Check Divisibility before Taking the Catalan Quotient

Use natural index n=23≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(23, 8233430727600, 24, 0, 343059613650, 343059613650, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (23, 8233430727600, 24, 0, 343059613650, 343059613650, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0049/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 50. Check Divisibility before Taking the Catalan Quotient

Use natural index n=24≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, central_binomial, denominator, remainder, quotient, binomial_difference, equal).

View answer and explanation(24, 32247603683100, 25, 0, 1289904147324, 1289904147324, Yes)

Compute binomial(2n,n), denominator n+1 and remainder first. Only remainder0 permits the exact quotient; independently compare binomial(2n,n)−binomial(2n,n+1). No integer truncation or shifted index is allowed. The exact result, retaining the requested order, is (24, 32247603683100, 25, 0, 1289904147324, 1289904147324, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0050/)

Catalan numbers · qf-mod42-catalan-closed-form · candidate





## 51. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=0≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(0, 1, 1, 1, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (0, 1, 1, 1, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0051/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 52. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=1≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(1, 1, 1, 1, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (1, 1, 1, 1, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0052/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 53. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=2≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(2, 2, 2, 2, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (2, 2, 2, 2, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0053/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 54. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=3≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(3, 5, 5, 5, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (3, 5, 5, 5, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0054/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 55. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=4≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(4, 14, 14, 14, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (4, 14, 14, 14, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0055/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 56. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=5≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(5, 42, 42, 42, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (5, 42, 42, 42, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0056/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 57. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=6≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(6, 132, 132, 132, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (6, 132, 132, 132, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0057/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 58. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=7≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(7, 429, 429, 429, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (7, 429, 429, 429, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0058/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 59. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=8≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(8, 1430, 1430, 1430, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (8, 1430, 1430, 1430, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0059/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 60. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=9≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(9, 4862, 4862, 4862, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (9, 4862, 4862, 4862, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0060/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 61. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=10≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(10, 16796, 16796, 16796, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (10, 16796, 16796, 16796, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0061/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 62. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=11≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(11, 58786, 58786, 58786, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (11, 58786, 58786, 58786, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0062/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 63. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=12≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(12, 208012, 208012, 208012, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (12, 208012, 208012, 208012, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0063/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 64. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=13≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(13, 742900, 742900, 742900, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (13, 742900, 742900, 742900, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0064/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 65. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=14≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(14, 2674440, 2674440, 2674440, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (14, 2674440, 2674440, 2674440, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0065/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 66. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=15≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(15, 9694845, 9694845, 9694845, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (15, 9694845, 9694845, 9694845, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0066/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 67. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=16≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(16, 35357670, 35357670, 35357670, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (16, 35357670, 35357670, 35357670, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0067/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 68. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=17≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(17, 129644790, 129644790, 129644790, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (17, 129644790, 129644790, 129644790, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0068/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 69. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=18≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(18, 477638700, 477638700, 477638700, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (18, 477638700, 477638700, 477638700, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0069/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 70. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=19≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(19, 1767263190, 1767263190, 1767263190, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (19, 1767263190, 1767263190, 1767263190, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0070/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 71. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=20≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(20, 6564120420, 6564120420, 6564120420, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (20, 6564120420, 6564120420, 6564120420, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0071/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 72. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=21≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(21, 24466267020, 24466267020, 24466267020, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (21, 24466267020, 24466267020, 24466267020, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0072/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 73. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=22≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(22, 91482563640, 91482563640, 91482563640, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (22, 91482563640, 91482563640, 91482563640, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0073/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 74. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=23≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(23, 343059613650, 343059613650, 343059613650, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (23, 343059613650, 343059613650, 343059613650, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0074/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 75. The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Use natural index n=24≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

View answer and explanation(24, 1289904147324, 1289904147324, 1289904147324, Yes, C_zero_index_convention_preserved)

Compare the convolution value, exact central-binomial quotient and adjacent-binomial difference at the same n, including n=0. These finite checks do not automatically prove every Catalan combinatorial bijection. The exact result, retaining the requested order, is (24, 1289904147324, 1289904147324, 1289904147324, Yes, C_zero_index_convention_preserved).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0075/)

Catalan numbers · qf-mod42-catalan-recurrence-closed-cross-check · candidate





## 76. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-1. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-1, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-1, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0076/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 77. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-2. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-2, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-2, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0077/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 78. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-3. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-3, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-3, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0078/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 79. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-4. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-4, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-4, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0079/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 80. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-5. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-5, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-5, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0080/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 81. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-6. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-6, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-6, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0081/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 82. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-7. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-7, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-7, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0082/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 83. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-8. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-8, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-8, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0083/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 84. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-9. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-9, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-9, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0084/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 85. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-10. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-10, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-10, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0085/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 86. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-11. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-11, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-11, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0086/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 87. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-12. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-12, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-12, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0087/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 88. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-13. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-13, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-13, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0088/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 89. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-14. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-14, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-14, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0089/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 90. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-15. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-15, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-15, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0090/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 91. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-16. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-16, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-16, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0091/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 92. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-17. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-17, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-17, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0092/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 93. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-18. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-18, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-18, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0093/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 94. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-19. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-19, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-19, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0094/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 95. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-20. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-20, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-20, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0095/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 96. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-21. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-21, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-21, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0096/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 97. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-22. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-22, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-22, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0097/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 98. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-23. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-23, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-23, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0098/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 99. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-24. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-24, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-24, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0099/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate





## 100. Negative Indices Are Outside This Catalan Domain

Use negative integer n=-25. Give (n, in_domain, value_or_null, boundary).

View answer and explanation(-25, No, null, Catalan_domain_is_natural_numbers_including_zero)

Report No natural-domain membership and null value; no negative-index extension is silently adopted. The exact result, retaining the requested order, is (-25, No, null, Catalan_domain_is_natural_numbers_including_zero).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261111-r05-catalan-numbers-0100/)

Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261111-r05-catalan-numbers-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/validation-report.json)