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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Catalan numbers · qf-mod42-catalan-negative-index-boundary · candidate
Canonical question JSONL · English JSONL · Validation report