# Question 22779: Catalan Generation Starts with C Zero Equal to One [Return to catalogue page 114](https://ocme.evemisslab.com/en/questions/page/114/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261111-r05-catalan-numbers-foundations/#q-20261014-advance-20261111-r05-catalan-numbers-0004) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0044-22779 · q-20261014-advance-20261111-r05-catalan-numbers-0004 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-catalan-numbers](https://ocme.evemisslab.com/en/mko/mko-catalan-numbers/) ## Step-by-step derivation S001 · impliesBefore source operation=catalan-recurrence-generation;original snapshot and mathematics retained After explicit ordered source output fields=[n, prefix, C_n, index_convention] Conditions: none Keep the historical interface marker and list ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their scope.The ordering helper supplies protocol order,not mathematical correctness. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore ORIGINAL requested natural index n=3 After initialize C_0=1;recurrence C_(j+1)=sum_i=0..j C_i*C_(j-i),including ALL splits Conditions: none There is one empty object,so C0=1.Include every split i0..j for C_(j+1),without a C1-based index shift. Step identifiers and dependenciesRule: catalan_base_case_empty_object_C0_one · Uses: G001, S001 S003 · calculationBefore Catalan complete convolution for C_1 := sum(1) After Catalan complete convolution for C_1 = 1 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore ORIGINAL index j=0;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1]] After C_1=1;ALL earlier values used from the established prefix=[1] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S003 S005 · calculationBefore Catalan complete convolution for C_2 := sum(1, 1) After Catalan complete convolution for C_2 = 2 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · impliesBefore ORIGINAL index j=1;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1, 1], [1, 1, 1, 1]] After C_2=2;ALL earlier values used from the established prefix=[1, 1] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S005 S007 · calculationBefore Catalan complete convolution for C_3 := sum(2, 1, 2) After Catalan complete convolution for C_3 = 5 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL index j=2;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 2, 2], [1, 1, 1, 1], [2, 2, 1, 2]] After C_3=5;ALL earlier values used from the established prefix=[1, 1, 2] Conditions: none List both established Catalan values and their exact product at every split.The complete sum gives the next term.Keep exact BigInt semantics even beyond SAFE,without truncation or floats. Step identifiers and dependenciesRule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S007 S009 · impliesBefore Catalan central:ORIGINAL binomial N=6,K=3 After use k=min(K,N-K)=3;initialize exact product1;empty product for k0 is1 Conditions: none Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations. Step identifiers and dependenciesRule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S008 S010 · calculationBefore Catalan central binomial product at i=1 := mul(1, 4) After Catalan central binomial product at i=1 = 4 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Catalan central exact binomial division by i=1 := div(4, 1) After Catalan central exact binomial division by i=1 = 4 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Catalan central binomial product at i=2 := mul(4, 5) After Catalan central binomial product at i=2 = 20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Catalan central exact binomial division by i=2 := div(20, 2) After Catalan central exact binomial division by i=2 = 10 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Catalan central binomial product at i=3 := mul(10, 6) After Catalan central binomial product at i=3 = 60 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Catalan central exact binomial division by i=3 := div(60, 3) After Catalan central exact binomial division by i=3 = 20 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Catalan ORIGINAL denominator n plus1 := sum(3, 1) After Catalan ORIGINAL denominator n plus1 = 4 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore ORIGINAL n=3;central binomial=20;positive denominator n+1=4 After Euclidean quotient=5,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer Conditions: none Check remainder0 before taking an exact integer quotient.Do not hide nondivisibility by truncation;retain denominator1 and central1 at n0. Step identifiers and dependenciesRule: catalan_remainder_BEFORE_exact_closed_division · Uses: G001, S016 S018 · calculationBefore Catalan exact closed quotient after remainder0 := div(20, 4) After Catalan exact closed quotient after remainder0 = 5 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore Catalan adjacent:ORIGINAL binomial N=6,K=4 After use k=min(K,N-K)=2;initialize exact product1;empty product for k0 is1 Conditions: none Use binomial symmetry to shorten the product while retaining original N/K roles.Multiply before each exact division,without floating-point factorial approximations. Step identifiers and dependenciesRule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S018 S020 · calculationBefore Catalan adjacent binomial product at i=1 := mul(1, 5) After Catalan adjacent binomial product at i=1 = 5 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore Catalan adjacent exact binomial division by i=1 := div(5, 1) After Catalan adjacent exact binomial division by i=1 = 5 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Catalan adjacent binomial product at i=2 := mul(5, 6) After Catalan adjacent binomial product at i=2 = 30 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Catalan adjacent exact binomial division by i=2 := div(30, 2) After Catalan adjacent exact binomial division by i=2 = 15 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Catalan central minus adjacent binomial := sub(20, 15) After Catalan central minus adjacent binomial = 5 Conditions: none Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · impliesBefore ORIGINAL n=3;recurrence value=5;central=20,adjacent=15,denominator=4 After closed quotient=5;binomial difference=5;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1) Conditions: none Reflect a bad balanced path through its first crossing of−1 to obtain a path with n+1 ups and n−1 downs,so good paths are the binomial difference.First-return decomposition U P D Q gives the Catalan convolution with empty-path base1,uniquely determining the same count.The adjacent-binomial ratio yields central/(n+1).This is a general argument,not extrapolation from finitely many equalities. Step identifiers and dependenciesRule: catalan_general_Dyck_reflection_and_first_return_recurrence · Uses: G001, S024 ## Reverse checks - S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S024 · premise_and_result_audit — Recompute Catalan central minus adjacent binomial with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=2 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=1 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=1 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Recompute Catalan exact closed quotient after remainder0 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Recompute Catalan ORIGINAL denominator n plus1 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=3 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Catalan central binomial product at i=3 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=2 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Catalan central binomial product at i=2 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=1 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Catalan central binomial product at i=1 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Recompute Catalan complete convolution for C_3 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Recompute Catalan complete convolution for C_2 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Recompute Catalan complete convolution for C_1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (3, [1, 1, 2, 5], 5, C_next_sum_i_zero_to_n_minus_one) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 8de45a723afa432d166dbf74120624a53cbffe2a58c4eca2f240b31a0605e32f - candidate record SHA-256: d7eefab2f86c95065fb041566942818ba9ad94b4fb27241db132ccbb46207880 - semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81 - question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261111-r05-catalan-numbers-0004.json)