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Question 22764: The Recurrence and Both Closed Forms Use the Same Zero-Based Index

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0044-22764 · q-20261014-advance-20261111-r05-catalan-numbers-0064

Original question and givens

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).

  • G001 — 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).

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.

Step-by-step derivation

S001 · implies

Before

source operation=catalan-recurrence-closed-cross-check;original snapshot and mathematics retained

After

explicit ordered source output fields=[n, recurrence_value, closed_value, difference_value, all_equal, certificate]

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 dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL requested natural index n=13

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 dependencies

Rule: catalan_base_case_empty_object_C0_one · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S007

S009 · calculation

Before

Catalan complete convolution for C_4 := sum(5, 2, 2, 5)

After

Catalan complete convolution for C_4 = 14

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

ORIGINAL index j=3;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 5, 5], [1, 1, 2, 2], [2, 2, 1, 2], [3, 5, 1, 5]]

After

C_4=14;ALL earlier values used from the established prefix=[1, 1, 2, 5]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S009

S011 · calculation

Before

Catalan complete convolution for C_5 := sum(14, 5, 4, 5, 14)

After

Catalan complete convolution for C_5 = 42

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · implies

Before

ORIGINAL index j=4;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 14, 14], [1, 1, 5, 5], [2, 2, 2, 4], [3, 5, 1, 5], [4, 14, 1, 14]]

After

C_5=42;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S011

S013 · calculation

Before

Catalan complete convolution for C_6 := sum(42, 14, 10, 10, 14, 42)

After

Catalan complete convolution for C_6 = 132

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · implies

Before

ORIGINAL index j=5;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 42, 42], [1, 1, 14, 14], [2, 2, 5, 10], [3, 5, 2, 10], [4, 14, 1, 14], [5, 42, 1, 42]]

After

C_6=132;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S013

S015 · calculation

Before

Catalan complete convolution for C_7 := sum(132, 42, 28, 25, 28, 42, 132)

After

Catalan complete convolution for C_7 = 429

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · implies

Before

ORIGINAL index j=6;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 132, 132], [1, 1, 42, 42], [2, 2, 14, 28], [3, 5, 5, 25], [4, 14, 2, 28], [5, 42, 1, 42], [6, 132, 1, 132]]

After

C_7=429;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S015

S017 · calculation

Before

Catalan complete convolution for C_8 := sum(429, 132, 84, 70, 70, 84, 132, 429)

After

Catalan complete convolution for C_8 = 1430

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · implies

Before

ORIGINAL index j=7;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 429, 429], [1, 1, 132, 132], [2, 2, 42, 84], [3, 5, 14, 70], [4, 14, 5, 70], [5, 42, 2, 84], [6, 132, 1, 132], [7, 429, 1, 429]]

After

C_8=1430;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S017

S019 · calculation

Before

Catalan complete convolution for C_9 := sum(1430, 429, 264, 210, 196, 210, 264, 429, 1430)

After

Catalan complete convolution for C_9 = 4862

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

ORIGINAL index j=8;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1430, 1430], [1, 1, 429, 429], [2, 2, 132, 264], [3, 5, 42, 210], [4, 14, 14, 196], [5, 42, 5, 210], [6, 132, 2, 264], [7, 429, 1, 429], [8, 1430, 1, 1430]]

After

C_9=4862;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S019

S021 · calculation

Before

Catalan complete convolution for C_10 := sum(4862, 1430, 858, 660, 588, 588, 660, 858, 1430, 4862)

After

Catalan complete convolution for C_10 = 16796

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · implies

Before

ORIGINAL index j=9;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 4862, 4862], [1, 1, 1430, 1430], [2, 2, 429, 858], [3, 5, 132, 660], [4, 14, 42, 588], [5, 42, 14, 588], [6, 132, 5, 660], [7, 429, 2, 858], [8, 1430, 1, 1430], [9, 4862, 1, 4862]]

After

C_10=16796;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S021

S023 · calculation

Before

Catalan complete convolution for C_11 := sum(16796, 4862, 2860, 2145, 1848, 1764, 1848, 2145, 2860, 4862, 16796)

After

Catalan complete convolution for C_11 = 58786

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · implies

Before

ORIGINAL index j=10;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 16796, 16796], [1, 1, 4862, 4862], [2, 2, 1430, 2860], [3, 5, 429, 2145], [4, 14, 132, 1848], [5, 42, 42, 1764], [6, 132, 14, 1848], [7, 429, 5, 2145], [8, 1430, 2, 2860], [9, 4862, 1, 4862], [10, 16796, 1, 16796]]

After

C_11=58786;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S023

S025 · calculation

Before

Catalan complete convolution for C_12 := sum(58786, 16796, 9724, 7150, 6006, 5544, 5544, 6006, 7150, 9724, 16796, 58786)

After

Catalan complete convolution for C_12 = 208012

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · implies

Before

ORIGINAL index j=11;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 58786, 58786], [1, 1, 16796, 16796], [2, 2, 4862, 9724], [3, 5, 1430, 7150], [4, 14, 429, 6006], [5, 42, 132, 5544], [6, 132, 42, 5544], [7, 429, 14, 6006], [8, 1430, 5, 7150], [9, 4862, 2, 9724], [10, 16796, 1, 16796], [11, 58786, 1, 58786]]

After

C_12=208012;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S025

S027 · calculation

Before

Catalan complete convolution for C_13 := sum(208012, 58786, 33592, 24310, 20020, 18018, 17424, 18018, 20020, 24310, 33592, 58786, 208012)

After

Catalan complete convolution for C_13 = 742900

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · implies

Before

ORIGINAL index j=12;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 208012, 208012], [1, 1, 58786, 58786], [2, 2, 16796, 33592], [3, 5, 4862, 24310], [4, 14, 1430, 20020], [5, 42, 429, 18018], [6, 132, 132, 17424], [7, 429, 42, 18018], [8, 1430, 14, 20020], [9, 4862, 5, 24310], [10, 16796, 2, 33592], [11, 58786, 1, 58786], [12, 208012, 1, 208012]]

After

C_13=742900;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012]

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 dependencies

Rule: catalan_EVERY_convolution_split_product_and_prefix · Uses: G001, S027

S029 · implies

Before

Catalan central:ORIGINAL binomial N=26,K=13

After

use k=min(K,N-K)=13;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 dependencies

Rule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S028

S030 · calculation

Before

Catalan central binomial product at i=1 := mul(1, 14)

After

Catalan central binomial product at i=1 = 14

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

Catalan central exact binomial division by i=1 := div(14, 1)

After

Catalan central exact binomial division by i=1 = 14

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

Catalan central binomial product at i=2 := mul(14, 15)

After

Catalan central binomial product at i=2 = 210

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

Catalan central exact binomial division by i=2 := div(210, 2)

After

Catalan central exact binomial division by i=2 = 105

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

Catalan central binomial product at i=3 := mul(105, 16)

After

Catalan central binomial product at i=3 = 1680

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

Catalan central exact binomial division by i=3 := div(1680, 3)

After

Catalan central exact binomial division by i=3 = 560

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

Catalan central binomial product at i=4 := mul(560, 17)

After

Catalan central binomial product at i=4 = 9520

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

Catalan central exact binomial division by i=4 := div(9520, 4)

After

Catalan central exact binomial division by i=4 = 2380

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

Catalan central binomial product at i=5 := mul(2380, 18)

After

Catalan central binomial product at i=5 = 42840

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

Catalan central exact binomial division by i=5 := div(42840, 5)

After

Catalan central exact binomial division by i=5 = 8568

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

Catalan central binomial product at i=6 := mul(8568, 19)

After

Catalan central binomial product at i=6 = 162792

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

Catalan central exact binomial division by i=6 := div(162792, 6)

After

Catalan central exact binomial division by i=6 = 27132

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

Catalan central binomial product at i=7 := mul(27132, 20)

After

Catalan central binomial product at i=7 = 542640

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

Catalan central exact binomial division by i=7 := div(542640, 7)

After

Catalan central exact binomial division by i=7 = 77520

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

Catalan central binomial product at i=8 := mul(77520, 21)

After

Catalan central binomial product at i=8 = 1627920

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

Catalan central exact binomial division by i=8 := div(1627920, 8)

After

Catalan central exact binomial division by i=8 = 203490

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

Catalan central binomial product at i=9 := mul(203490, 22)

After

Catalan central binomial product at i=9 = 4476780

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

Catalan central exact binomial division by i=9 := div(4476780, 9)

After

Catalan central exact binomial division by i=9 = 497420

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

Catalan central binomial product at i=10 := mul(497420, 23)

After

Catalan central binomial product at i=10 = 11440660

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

Catalan central exact binomial division by i=10 := div(11440660, 10)

After

Catalan central exact binomial division by i=10 = 1144066

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

Catalan central binomial product at i=11 := mul(1144066, 24)

After

Catalan central binomial product at i=11 = 27457584

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

Catalan central exact binomial division by i=11 := div(27457584, 11)

After

Catalan central exact binomial division by i=11 = 2496144

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

Catalan central binomial product at i=12 := mul(2496144, 25)

After

Catalan central binomial product at i=12 = 62403600

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · calculation

Before

Catalan central exact binomial division by i=12 := div(62403600, 12)

After

Catalan central exact binomial division by i=12 = 5200300

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S052

S054 · calculation

Before

Catalan central binomial product at i=13 := mul(5200300, 26)

After

Catalan central binomial product at i=13 = 135207800

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · calculation

Before

Catalan central exact binomial division by i=13 := div(135207800, 13)

After

Catalan central exact binomial division by i=13 = 10400600

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S054

S056 · calculation

Before

Catalan ORIGINAL denominator n plus1 := sum(13, 1)

After

Catalan ORIGINAL denominator n plus1 = 14

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S055

S057 · implies

Before

ORIGINAL n=13;central binomial=10400600;positive denominator n+1=14

After

Euclidean quotient=742900,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 dependencies

Rule: catalan_remainder_BEFORE_exact_closed_division · Uses: G001, S056

S058 · calculation

Before

Catalan exact closed quotient after remainder0 := div(10400600, 14)

After

Catalan exact closed quotient after remainder0 = 742900

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S057

S059 · implies

Before

Catalan adjacent:ORIGINAL binomial N=26,K=14

After

use k=min(K,N-K)=12;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 dependencies

Rule: catalan_exact_binomial_product_initial_and_symmetry · Uses: G001, S058

S060 · calculation

Before

Catalan adjacent binomial product at i=1 := mul(1, 15)

After

Catalan adjacent binomial product at i=1 = 15

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S059

S061 · calculation

Before

Catalan adjacent exact binomial division by i=1 := div(15, 1)

After

Catalan adjacent exact binomial division by i=1 = 15

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S060

S062 · calculation

Before

Catalan adjacent binomial product at i=2 := mul(15, 16)

After

Catalan adjacent binomial product at i=2 = 240

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S061

S063 · calculation

Before

Catalan adjacent exact binomial division by i=2 := div(240, 2)

After

Catalan adjacent exact binomial division by i=2 = 120

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S062

S064 · calculation

Before

Catalan adjacent binomial product at i=3 := mul(120, 17)

After

Catalan adjacent binomial product at i=3 = 2040

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S063

S065 · calculation

Before

Catalan adjacent exact binomial division by i=3 := div(2040, 3)

After

Catalan adjacent exact binomial division by i=3 = 680

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S064

S066 · calculation

Before

Catalan adjacent binomial product at i=4 := mul(680, 18)

After

Catalan adjacent binomial product at i=4 = 12240

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S065

S067 · calculation

Before

Catalan adjacent exact binomial division by i=4 := div(12240, 4)

After

Catalan adjacent exact binomial division by i=4 = 3060

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S066

S068 · calculation

Before

Catalan adjacent binomial product at i=5 := mul(3060, 19)

After

Catalan adjacent binomial product at i=5 = 58140

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S067

S069 · calculation

Before

Catalan adjacent exact binomial division by i=5 := div(58140, 5)

After

Catalan adjacent exact binomial division by i=5 = 11628

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S068

S070 · calculation

Before

Catalan adjacent binomial product at i=6 := mul(11628, 20)

After

Catalan adjacent binomial product at i=6 = 232560

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S069

S071 · calculation

Before

Catalan adjacent exact binomial division by i=6 := div(232560, 6)

After

Catalan adjacent exact binomial division by i=6 = 38760

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S070

S072 · calculation

Before

Catalan adjacent binomial product at i=7 := mul(38760, 21)

After

Catalan adjacent binomial product at i=7 = 813960

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S071

S073 · calculation

Before

Catalan adjacent exact binomial division by i=7 := div(813960, 7)

After

Catalan adjacent exact binomial division by i=7 = 116280

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S072

S074 · calculation

Before

Catalan adjacent binomial product at i=8 := mul(116280, 22)

After

Catalan adjacent binomial product at i=8 = 2558160

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S073

S075 · calculation

Before

Catalan adjacent exact binomial division by i=8 := div(2558160, 8)

After

Catalan adjacent exact binomial division by i=8 = 319770

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S074

S076 · calculation

Before

Catalan adjacent binomial product at i=9 := mul(319770, 23)

After

Catalan adjacent binomial product at i=9 = 7354710

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S075

S077 · calculation

Before

Catalan adjacent exact binomial division by i=9 := div(7354710, 9)

After

Catalan adjacent exact binomial division by i=9 = 817190

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S076

S078 · calculation

Before

Catalan adjacent binomial product at i=10 := mul(817190, 24)

After

Catalan adjacent binomial product at i=10 = 19612560

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S077

S079 · calculation

Before

Catalan adjacent exact binomial division by i=10 := div(19612560, 10)

After

Catalan adjacent exact binomial division by i=10 = 1961256

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S078

S080 · calculation

Before

Catalan adjacent binomial product at i=11 := mul(1961256, 25)

After

Catalan adjacent binomial product at i=11 = 49031400

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S079

S081 · calculation

Before

Catalan adjacent exact binomial division by i=11 := div(49031400, 11)

After

Catalan adjacent exact binomial division by i=11 = 4457400

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S080

S082 · calculation

Before

Catalan adjacent binomial product at i=12 := mul(4457400, 26)

After

Catalan adjacent binomial product at i=12 = 115892400

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S081

S083 · calculation

Before

Catalan adjacent exact binomial division by i=12 := div(115892400, 12)

After

Catalan adjacent exact binomial division by i=12 = 9657700

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S082

S084 · calculation

Before

Catalan central minus adjacent binomial := sub(10400600, 9657700)

After

Catalan central minus adjacent binomial = 742900

Conditions: none

Substitute original roles,indices and established intermediates exactly,not floating-point approximations or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S083

S085 · implies

Before

ORIGINAL n=13;recurrence value=742900;central=10400600,adjacent=9657700,denominator=14

After

closed quotient=742900;binomial difference=742900;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 dependencies

Rule: catalan_general_Dyck_reflection_and_first_return_recurrence · Uses: G001, S084

Reverse checks

  • S085 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S084 · premise_and_result_audit — Recompute Catalan central minus adjacent binomial with exact rational arithmetic, not rounded decimals.
  • S083 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=12 with exact rational arithmetic, not rounded decimals.
  • S082 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=12 with exact rational arithmetic, not rounded decimals.
  • S081 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=11 with exact rational arithmetic, not rounded decimals.
  • S080 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=11 with exact rational arithmetic, not rounded decimals.
  • S079 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=10 with exact rational arithmetic, not rounded decimals.
  • S078 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=10 with exact rational arithmetic, not rounded decimals.
  • S077 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=9 with exact rational arithmetic, not rounded decimals.
  • S076 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=9 with exact rational arithmetic, not rounded decimals.
  • S075 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=8 with exact rational arithmetic, not rounded decimals.
  • S074 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=8 with exact rational arithmetic, not rounded decimals.
  • S073 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=7 with exact rational arithmetic, not rounded decimals.
  • S072 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=7 with exact rational arithmetic, not rounded decimals.
  • S071 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=6 with exact rational arithmetic, not rounded decimals.
  • S070 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=6 with exact rational arithmetic, not rounded decimals.
  • S069 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=5 with exact rational arithmetic, not rounded decimals.
  • S068 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=5 with exact rational arithmetic, not rounded decimals.
  • S067 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=4 with exact rational arithmetic, not rounded decimals.
  • S066 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=4 with exact rational arithmetic, not rounded decimals.
  • S065 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=3 with exact rational arithmetic, not rounded decimals.
  • S064 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=3 with exact rational arithmetic, not rounded decimals.
  • S063 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=2 with exact rational arithmetic, not rounded decimals.
  • S062 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=2 with exact rational arithmetic, not rounded decimals.
  • S061 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=1 with exact rational arithmetic, not rounded decimals.
  • S060 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=1 with exact rational arithmetic, not rounded decimals.
  • S059 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S058 · premise_and_result_audit — Recompute Catalan exact closed quotient after remainder0 with exact rational arithmetic, not rounded decimals.
  • S057 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S056 · premise_and_result_audit — Recompute Catalan ORIGINAL denominator n plus1 with exact rational arithmetic, not rounded decimals.
  • S055 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=13 with exact rational arithmetic, not rounded decimals.
  • S054 · premise_and_result_audit — Recompute Catalan central binomial product at i=13 with exact rational arithmetic, not rounded decimals.
  • S053 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=12 with exact rational arithmetic, not rounded decimals.
  • S052 · premise_and_result_audit — Recompute Catalan central binomial product at i=12 with exact rational arithmetic, not rounded decimals.
  • S051 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=11 with exact rational arithmetic, not rounded decimals.
  • S050 · premise_and_result_audit — Recompute Catalan central binomial product at i=11 with exact rational arithmetic, not rounded decimals.
  • S049 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=10 with exact rational arithmetic, not rounded decimals.
  • S048 · premise_and_result_audit — Recompute Catalan central binomial product at i=10 with exact rational arithmetic, not rounded decimals.
  • S047 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=9 with exact rational arithmetic, not rounded decimals.
  • S046 · premise_and_result_audit — Recompute Catalan central binomial product at i=9 with exact rational arithmetic, not rounded decimals.
  • S045 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=8 with exact rational arithmetic, not rounded decimals.
  • S044 · premise_and_result_audit — Recompute Catalan central binomial product at i=8 with exact rational arithmetic, not rounded decimals.
  • S043 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=7 with exact rational arithmetic, not rounded decimals.
  • S042 · premise_and_result_audit — Recompute Catalan central binomial product at i=7 with exact rational arithmetic, not rounded decimals.
  • S041 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=6 with exact rational arithmetic, not rounded decimals.
  • S040 · premise_and_result_audit — Recompute Catalan central binomial product at i=6 with exact rational arithmetic, not rounded decimals.
  • S039 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=5 with exact rational arithmetic, not rounded decimals.
  • S038 · premise_and_result_audit — Recompute Catalan central binomial product at i=5 with exact rational arithmetic, not rounded decimals.
  • S037 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=4 with exact rational arithmetic, not rounded decimals.
  • S036 · premise_and_result_audit — Recompute Catalan central binomial product at i=4 with exact rational arithmetic, not rounded decimals.
  • S035 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=3 with exact rational arithmetic, not rounded decimals.
  • S034 · premise_and_result_audit — Recompute Catalan central binomial product at i=3 with exact rational arithmetic, not rounded decimals.
  • S033 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=2 with exact rational arithmetic, not rounded decimals.
  • S032 · premise_and_result_audit — Recompute Catalan central binomial product at i=2 with exact rational arithmetic, not rounded decimals.
  • S031 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=1 with exact rational arithmetic, not rounded decimals.
  • S030 · premise_and_result_audit — Recompute Catalan central binomial product at i=1 with exact rational arithmetic, not rounded decimals.
  • S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S028 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S027 · premise_and_result_audit — Recompute Catalan complete convolution for C_13 with exact rational arithmetic, not rounded decimals.
  • S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S025 · premise_and_result_audit — Recompute Catalan complete convolution for C_12 with exact rational arithmetic, not rounded decimals.
  • S024 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S023 · premise_and_result_audit — Recompute Catalan complete convolution for C_11 with exact rational arithmetic, not rounded decimals.
  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Recompute Catalan complete convolution for C_10 with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Recompute Catalan complete convolution for C_9 with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Recompute Catalan complete convolution for C_8 with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Recompute Catalan complete convolution for C_7 with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Recompute Catalan complete convolution for C_6 with exact rational arithmetic, not rounded decimals.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Recompute Catalan complete convolution for C_5 with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S009 · premise_and_result_audit — Recompute Catalan complete convolution for C_4 with exact rational arithmetic, not rounded decimals.
  • 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

(13, 742900, 742900, 742900, Yes, C_zero_index_convention_preserved)

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: c71606dc043fa021b3f451e85b4b998eb55f65d17712b859e9279c04efa2ce19
  • candidate record SHA-256: fdc34867bd665f60169d498614d9998c2caf6ab0817c0c8f566f99e68b8b8bbe
  • semantic review receipt SHA-256: 89b8e810eced1a69dbfef28d4b112e829edf156c7905a86e39a3f4958cd75e81
  • question-first witness SHA-256: b2e8b769c463242f05172564ae96e310403eae151e1ec30bcb50be3ffe12cfc7
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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