Question 22722: Check Divisibility before Taking the Catalan Quotient

Original question and givens

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).

• G001 — 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).

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

Step-by-step derivation

source operation=catalan-closed-form;original snapshot and mathematics retained

explicit ordered source output fields=[n, central_binomial, denominator, remainder, quotient, binomial_difference, equal]

ORIGINAL requested natural index n=21

initialize C_0=1;recurrence C_(j+1)=sum_i=0..j C_i*C_(j-i),including ALL splits

Catalan complete convolution for C_1 := sum(1)

Catalan complete convolution for C_1 = 1

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

C_1=1;ALL earlier values used from the established prefix=[1]

Catalan complete convolution for C_2 := sum(1, 1)

Catalan complete convolution for C_2 = 2

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

C_2=2;ALL earlier values used from the established prefix=[1, 1]

Catalan complete convolution for C_3 := sum(2, 1, 2)

Catalan complete convolution for C_3 = 5

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]]

C_3=5;ALL earlier values used from the established prefix=[1, 1, 2]

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

Catalan complete convolution for C_4 = 14

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]]

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

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

Catalan complete convolution for C_5 = 42

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]]

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

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

Catalan complete convolution for C_6 = 132

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]]

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

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

Catalan complete convolution for C_7 = 429

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]]

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

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

Catalan complete convolution for C_8 = 1430

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]]

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

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

Catalan complete convolution for C_9 = 4862

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]]

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

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

Catalan complete convolution for C_10 = 16796

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]]

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

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

Catalan complete convolution for C_11 = 58786

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]]

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

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

Catalan complete convolution for C_12 = 208012

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]]

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

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

Catalan complete convolution for C_13 = 742900

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]]

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

Catalan complete convolution for C_14 := sum(742900, 208012, 117572, 83980, 68068, 60060, 56628, 56628, 60060, 68068, 83980, 117572, 208012, 742900)

Catalan complete convolution for C_14 = 2674440

ORIGINAL index j=13;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 742900, 742900], [1, 1, 208012, 208012], [2, 2, 58786, 117572], [3, 5, 16796, 83980], [4, 14, 4862, 68068], [5, 42, 1430, 60060], [6, 132, 429, 56628], [7, 429, 132, 56628], [8, 1430, 42, 60060], [9, 4862, 14, 68068], [10, 16796, 5, 83980], [11, 58786, 2, 117572], [12, 208012, 1, 208012], [13, 742900, 1, 742900]]

C_14=2674440;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900]

Catalan complete convolution for C_15 := sum(2674440, 742900, 416024, 293930, 235144, 204204, 188760, 184041, 188760, 204204, 235144, 293930, 416024, 742900, 2674440)

Catalan complete convolution for C_15 = 9694845

ORIGINAL index j=14;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 2674440, 2674440], [1, 1, 742900, 742900], [2, 2, 208012, 416024], [3, 5, 58786, 293930], [4, 14, 16796, 235144], [5, 42, 4862, 204204], [6, 132, 1430, 188760], [7, 429, 429, 184041], [8, 1430, 132, 188760], [9, 4862, 42, 204204], [10, 16796, 14, 235144], [11, 58786, 5, 293930], [12, 208012, 2, 416024], [13, 742900, 1, 742900], [14, 2674440, 1, 2674440]]

C_15=9694845;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440]

Catalan complete convolution for C_16 := sum(9694845, 2674440, 1485800, 1040060, 823004, 705432, 641784, 613470, 613470, 641784, 705432, 823004, 1040060, 1485800, 2674440, 9694845)

Catalan complete convolution for C_16 = 35357670

ORIGINAL index j=15;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 9694845, 9694845], [1, 1, 2674440, 2674440], [2, 2, 742900, 1485800], [3, 5, 208012, 1040060], [4, 14, 58786, 823004], [5, 42, 16796, 705432], [6, 132, 4862, 641784], [7, 429, 1430, 613470], [8, 1430, 429, 613470], [9, 4862, 132, 641784], [10, 16796, 42, 705432], [11, 58786, 14, 823004], [12, 208012, 5, 1040060], [13, 742900, 2, 1485800], [14, 2674440, 1, 2674440], [15, 9694845, 1, 9694845]]

C_16=35357670;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845]

Catalan complete convolution for C_17 := sum(35357670, 9694845, 5348880, 3714500, 2912168, 2469012, 2217072, 2085798, 2044900, 2085798, 2217072, 2469012, 2912168, 3714500, 5348880, 9694845, 35357670)

Catalan complete convolution for C_17 = 129644790

ORIGINAL index j=16;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 35357670, 35357670], [1, 1, 9694845, 9694845], [2, 2, 2674440, 5348880], [3, 5, 742900, 3714500], [4, 14, 208012, 2912168], [5, 42, 58786, 2469012], [6, 132, 16796, 2217072], [7, 429, 4862, 2085798], [8, 1430, 1430, 2044900], [9, 4862, 429, 2085798], [10, 16796, 132, 2217072], [11, 58786, 42, 2469012], [12, 208012, 14, 2912168], [13, 742900, 5, 3714500], [14, 2674440, 2, 5348880], [15, 9694845, 1, 9694845], [16, 35357670, 1, 35357670]]

C_17=129644790;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670]

Catalan complete convolution for C_18 := sum(129644790, 35357670, 19389690, 13372200, 10400600, 8736504, 7759752, 7205484, 6952660, 6952660, 7205484, 7759752, 8736504, 10400600, 13372200, 19389690, 35357670, 129644790)

Catalan complete convolution for C_18 = 477638700

ORIGINAL index j=17;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 129644790, 129644790], [1, 1, 35357670, 35357670], [2, 2, 9694845, 19389690], [3, 5, 2674440, 13372200], [4, 14, 742900, 10400600], [5, 42, 208012, 8736504], [6, 132, 58786, 7759752], [7, 429, 16796, 7205484], [8, 1430, 4862, 6952660], [9, 4862, 1430, 6952660], [10, 16796, 429, 7205484], [11, 58786, 132, 7759752], [12, 208012, 42, 8736504], [13, 742900, 14, 10400600], [14, 2674440, 5, 13372200], [15, 9694845, 2, 19389690], [16, 35357670, 1, 35357670], [17, 129644790, 1, 129644790]]

C_18=477638700;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790]

Catalan complete convolution for C_19 := sum(477638700, 129644790, 70715340, 48474225, 37442160, 31201800, 27457584, 25219194, 24018280, 23639044, 24018280, 25219194, 27457584, 31201800, 37442160, 48474225, 70715340, 129644790, 477638700)

Catalan complete convolution for C_19 = 1767263190

ORIGINAL index j=18;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 477638700, 477638700], [1, 1, 129644790, 129644790], [2, 2, 35357670, 70715340], [3, 5, 9694845, 48474225], [4, 14, 2674440, 37442160], [5, 42, 742900, 31201800], [6, 132, 208012, 27457584], [7, 429, 58786, 25219194], [8, 1430, 16796, 24018280], [9, 4862, 4862, 23639044], [10, 16796, 1430, 24018280], [11, 58786, 429, 25219194], [12, 208012, 132, 27457584], [13, 742900, 42, 31201800], [14, 2674440, 14, 37442160], [15, 9694845, 5, 48474225], [16, 35357670, 2, 70715340], [17, 129644790, 1, 129644790], [18, 477638700, 1, 477638700]]

C_19=1767263190;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700]

Catalan complete convolution for C_20 := sum(1767263190, 477638700, 259289580, 176788350, 135727830, 112326480, 98062800, 89237148, 84063980, 81662152, 81662152, 84063980, 89237148, 98062800, 112326480, 135727830, 176788350, 259289580, 477638700, 1767263190)

Catalan complete convolution for C_20 = 6564120420

ORIGINAL index j=19;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 1767263190, 1767263190], [1, 1, 477638700, 477638700], [2, 2, 129644790, 259289580], [3, 5, 35357670, 176788350], [4, 14, 9694845, 135727830], [5, 42, 2674440, 112326480], [6, 132, 742900, 98062800], [7, 429, 208012, 89237148], [8, 1430, 58786, 84063980], [9, 4862, 16796, 81662152], [10, 16796, 4862, 81662152], [11, 58786, 1430, 84063980], [12, 208012, 429, 89237148], [13, 742900, 132, 98062800], [14, 2674440, 42, 112326480], [15, 9694845, 14, 135727830], [16, 35357670, 5, 176788350], [17, 129644790, 2, 259289580], [18, 477638700, 1, 477638700], [19, 1767263190, 1, 1767263190]]

C_20=6564120420;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190]

Catalan complete convolution for C_21 := sum(6564120420, 1767263190, 955277400, 648223950, 495007380, 407183490, 353026080, 318704100, 297457160, 285817532, 282105616, 285817532, 297457160, 318704100, 353026080, 407183490, 495007380, 648223950, 955277400, 1767263190, 6564120420)

Catalan complete convolution for C_21 = 24466267020

ORIGINAL index j=20;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 6564120420, 6564120420], [1, 1, 1767263190, 1767263190], [2, 2, 477638700, 955277400], [3, 5, 129644790, 648223950], [4, 14, 35357670, 495007380], [5, 42, 9694845, 407183490], [6, 132, 2674440, 353026080], [7, 429, 742900, 318704100], [8, 1430, 208012, 297457160], [9, 4862, 58786, 285817532], [10, 16796, 16796, 282105616], [11, 58786, 4862, 285817532], [12, 208012, 1430, 297457160], [13, 742900, 429, 318704100], [14, 2674440, 132, 353026080], [15, 9694845, 42, 407183490], [16, 35357670, 14, 495007380], [17, 129644790, 5, 648223950], [18, 477638700, 2, 955277400], [19, 1767263190, 1, 1767263190], [20, 6564120420, 1, 6564120420]]

C_21=24466267020;ALL earlier values used from the established prefix=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420]

Catalan central:ORIGINAL binomial N=42,K=21

use k=min(K,N-K)=21;initialize exact product1;empty product for k0 is1

Catalan central binomial product at i=1 := mul(1, 22)

Catalan central binomial product at i=1 = 22

Catalan central exact binomial division by i=1 := div(22, 1)

Catalan central exact binomial division by i=1 = 22

Catalan central binomial product at i=2 := mul(22, 23)

Catalan central binomial product at i=2 = 506

Catalan central exact binomial division by i=2 := div(506, 2)

Catalan central exact binomial division by i=2 = 253

Catalan central binomial product at i=3 := mul(253, 24)

Catalan central binomial product at i=3 = 6072

Catalan central exact binomial division by i=3 := div(6072, 3)

Catalan central exact binomial division by i=3 = 2024

Catalan central binomial product at i=4 := mul(2024, 25)

Catalan central binomial product at i=4 = 50600

Catalan central exact binomial division by i=4 := div(50600, 4)

Catalan central exact binomial division by i=4 = 12650

Catalan central binomial product at i=5 := mul(12650, 26)

Catalan central binomial product at i=5 = 328900

Catalan central exact binomial division by i=5 := div(328900, 5)

Catalan central exact binomial division by i=5 = 65780

Catalan central binomial product at i=6 := mul(65780, 27)

Catalan central binomial product at i=6 = 1776060

Catalan central exact binomial division by i=6 := div(1776060, 6)

Catalan central exact binomial division by i=6 = 296010

Catalan central binomial product at i=7 := mul(296010, 28)

Catalan central binomial product at i=7 = 8288280

Catalan central exact binomial division by i=7 := div(8288280, 7)

Catalan central exact binomial division by i=7 = 1184040

Catalan central binomial product at i=8 := mul(1184040, 29)

Catalan central binomial product at i=8 = 34337160

Catalan central exact binomial division by i=8 := div(34337160, 8)

Catalan central exact binomial division by i=8 = 4292145

Catalan central binomial product at i=9 := mul(4292145, 30)

Catalan central binomial product at i=9 = 128764350

Catalan central exact binomial division by i=9 := div(128764350, 9)

Catalan central exact binomial division by i=9 = 14307150

Catalan central binomial product at i=10 := mul(14307150, 31)

Catalan central binomial product at i=10 = 443521650

Catalan central exact binomial division by i=10 := div(443521650, 10)

Catalan central exact binomial division by i=10 = 44352165

Catalan central binomial product at i=11 := mul(44352165, 32)

Catalan central binomial product at i=11 = 1419269280

Catalan central exact binomial division by i=11 := div(1419269280, 11)

Catalan central exact binomial division by i=11 = 129024480

Catalan central binomial product at i=12 := mul(129024480, 33)

Catalan central binomial product at i=12 = 4257807840

Catalan central exact binomial division by i=12 := div(4257807840, 12)

Catalan central exact binomial division by i=12 = 354817320

Catalan central binomial product at i=13 := mul(354817320, 34)

Catalan central binomial product at i=13 = 12063788880

Catalan central exact binomial division by i=13 := div(12063788880, 13)

Catalan central exact binomial division by i=13 = 927983760

Catalan central binomial product at i=14 := mul(927983760, 35)

Catalan central binomial product at i=14 = 32479431600

Catalan central exact binomial division by i=14 := div(32479431600, 14)

Catalan central exact binomial division by i=14 = 2319959400

Catalan central binomial product at i=15 := mul(2319959400, 36)

Catalan central binomial product at i=15 = 83518538400

Catalan central exact binomial division by i=15 := div(83518538400, 15)

Catalan central exact binomial division by i=15 = 5567902560

Catalan central binomial product at i=16 := mul(5567902560, 37)

Catalan central binomial product at i=16 = 206012394720

Catalan central exact binomial division by i=16 := div(206012394720, 16)

Catalan central exact binomial division by i=16 = 12875774670

Catalan central binomial product at i=17 := mul(12875774670, 38)

Catalan central binomial product at i=17 = 489279437460

Catalan central exact binomial division by i=17 := div(489279437460, 17)

Catalan central exact binomial division by i=17 = 28781143380

Catalan central binomial product at i=18 := mul(28781143380, 39)

Catalan central binomial product at i=18 = 1122464591820

Catalan central exact binomial division by i=18 := div(1122464591820, 18)

Catalan central exact binomial division by i=18 = 62359143990

Catalan central binomial product at i=19 := mul(62359143990, 40)

Catalan central binomial product at i=19 = 2494365759600

Catalan central exact binomial division by i=19 := div(2494365759600, 19)

Catalan central exact binomial division by i=19 = 131282408400

Catalan central binomial product at i=20 := mul(131282408400, 41)

Catalan central binomial product at i=20 = 5382578744400

Catalan central exact binomial division by i=20 := div(5382578744400, 20)

Catalan central exact binomial division by i=20 = 269128937220

Catalan central binomial product at i=21 := mul(269128937220, 42)

Catalan central binomial product at i=21 = 11303415363240

Catalan central exact binomial division by i=21 := div(11303415363240, 21)

Catalan central exact binomial division by i=21 = 538257874440

Catalan ORIGINAL denominator n plus1 := sum(21, 1)

Catalan ORIGINAL denominator n plus1 = 22

ORIGINAL n=21;central binomial=538257874440;positive denominator n+1=22

Euclidean quotient=24466267020,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer

Catalan exact closed quotient after remainder0 := div(538257874440, 22)

Catalan exact closed quotient after remainder0 = 24466267020

Catalan adjacent:ORIGINAL binomial N=42,K=22

use k=min(K,N-K)=20;initialize exact product1;empty product for k0 is1

Catalan adjacent binomial product at i=1 := mul(1, 23)

Catalan adjacent binomial product at i=1 = 23

Catalan adjacent exact binomial division by i=1 := div(23, 1)

Catalan adjacent exact binomial division by i=1 = 23

Catalan adjacent binomial product at i=2 := mul(23, 24)

Catalan adjacent binomial product at i=2 = 552

Catalan adjacent exact binomial division by i=2 := div(552, 2)

Catalan adjacent exact binomial division by i=2 = 276

Catalan adjacent binomial product at i=3 := mul(276, 25)

Catalan adjacent binomial product at i=3 = 6900

Catalan adjacent exact binomial division by i=3 := div(6900, 3)

Catalan adjacent exact binomial division by i=3 = 2300

Catalan adjacent binomial product at i=4 := mul(2300, 26)

Catalan adjacent binomial product at i=4 = 59800

Catalan adjacent exact binomial division by i=4 := div(59800, 4)

Catalan adjacent exact binomial division by i=4 = 14950

Catalan adjacent binomial product at i=5 := mul(14950, 27)

Catalan adjacent binomial product at i=5 = 403650

Catalan adjacent exact binomial division by i=5 := div(403650, 5)

Catalan adjacent exact binomial division by i=5 = 80730

Catalan adjacent binomial product at i=6 := mul(80730, 28)

Catalan adjacent binomial product at i=6 = 2260440

Catalan adjacent exact binomial division by i=6 := div(2260440, 6)

Catalan adjacent exact binomial division by i=6 = 376740

Catalan adjacent binomial product at i=7 := mul(376740, 29)

Catalan adjacent binomial product at i=7 = 10925460

Catalan adjacent exact binomial division by i=7 := div(10925460, 7)

Catalan adjacent exact binomial division by i=7 = 1560780

Catalan adjacent binomial product at i=8 := mul(1560780, 30)

Catalan adjacent binomial product at i=8 = 46823400

Catalan adjacent exact binomial division by i=8 := div(46823400, 8)

Catalan adjacent exact binomial division by i=8 = 5852925

Catalan adjacent binomial product at i=9 := mul(5852925, 31)

Catalan adjacent binomial product at i=9 = 181440675

Catalan adjacent exact binomial division by i=9 := div(181440675, 9)

Catalan adjacent exact binomial division by i=9 = 20160075

Catalan adjacent binomial product at i=10 := mul(20160075, 32)

Catalan adjacent binomial product at i=10 = 645122400

Catalan adjacent exact binomial division by i=10 := div(645122400, 10)

Catalan adjacent exact binomial division by i=10 = 64512240

Catalan adjacent binomial product at i=11 := mul(64512240, 33)

Catalan adjacent binomial product at i=11 = 2128903920

Catalan adjacent exact binomial division by i=11 := div(2128903920, 11)

Catalan adjacent exact binomial division by i=11 = 193536720

Catalan adjacent binomial product at i=12 := mul(193536720, 34)

Catalan adjacent binomial product at i=12 = 6580248480

Catalan adjacent exact binomial division by i=12 := div(6580248480, 12)

Catalan adjacent exact binomial division by i=12 = 548354040

Catalan adjacent binomial product at i=13 := mul(548354040, 35)

Catalan adjacent binomial product at i=13 = 19192391400

Catalan adjacent exact binomial division by i=13 := div(19192391400, 13)

Catalan adjacent exact binomial division by i=13 = 1476337800

Catalan adjacent binomial product at i=14 := mul(1476337800, 36)

Catalan adjacent binomial product at i=14 = 53148160800

Catalan adjacent exact binomial division by i=14 := div(53148160800, 14)

Catalan adjacent exact binomial division by i=14 = 3796297200

Catalan adjacent binomial product at i=15 := mul(3796297200, 37)

Catalan adjacent binomial product at i=15 = 140462996400

Catalan adjacent exact binomial division by i=15 := div(140462996400, 15)

Catalan adjacent exact binomial division by i=15 = 9364199760

Catalan adjacent binomial product at i=16 := mul(9364199760, 38)

Catalan adjacent binomial product at i=16 = 355839590880

Catalan adjacent exact binomial division by i=16 := div(355839590880, 16)

Catalan adjacent exact binomial division by i=16 = 22239974430

Catalan adjacent binomial product at i=17 := mul(22239974430, 39)

Catalan adjacent binomial product at i=17 = 867359002770

Catalan adjacent exact binomial division by i=17 := div(867359002770, 17)

Catalan adjacent exact binomial division by i=17 = 51021117810

Catalan adjacent binomial product at i=18 := mul(51021117810, 40)

Catalan adjacent binomial product at i=18 = 2040844712400

Catalan adjacent exact binomial division by i=18 := div(2040844712400, 18)

Catalan adjacent exact binomial division by i=18 = 113380261800

Catalan adjacent binomial product at i=19 := mul(113380261800, 41)

Catalan adjacent binomial product at i=19 = 4648590733800

Catalan adjacent exact binomial division by i=19 := div(4648590733800, 19)

Catalan adjacent exact binomial division by i=19 = 244662670200

Catalan adjacent binomial product at i=20 := mul(244662670200, 42)

Catalan adjacent binomial product at i=20 = 10275832148400

Catalan adjacent exact binomial division by i=20 := div(10275832148400, 20)

Catalan adjacent exact binomial division by i=20 = 513791607420

Catalan central minus adjacent binomial := sub(538257874440, 513791607420)

Catalan central minus adjacent binomial = 24466267020

ORIGINAL n=21;recurrence value=24466267020;central=538257874440,adjacent=513791607420,denominator=22

closed quotient=24466267020;binomial difference=24466267020;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1)

Reverse checks

• S133 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S132 · premise_and_result_audit — Recompute Catalan central minus adjacent binomial with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=20 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=20 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=19 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=19 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=18 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=18 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=17 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=17 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=16 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=16 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=15 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=15 with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=14 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=14 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=13 with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=13 with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=12 with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=12 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=11 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=11 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=10 with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=10 with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=9 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=9 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=8 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=8 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=7 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=7 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=6 with exact rational arithmetic, not rounded decimals.
• S102 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=6 with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=5 with exact rational arithmetic, not rounded decimals.
• S100 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=5 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=4 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=4 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=3 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=3 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=2 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=2 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=1 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=1 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S090 · premise_and_result_audit — Recompute Catalan exact closed quotient after remainder0 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S088 · premise_and_result_audit — Recompute Catalan ORIGINAL denominator n plus1 with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=21 with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Catalan central binomial product at i=21 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=20 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Catalan central binomial product at i=20 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=19 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Catalan central binomial product at i=19 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=18 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Catalan central binomial product at i=18 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=17 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Catalan central binomial product at i=17 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=16 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Catalan central binomial product at i=16 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=15 with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Catalan central binomial product at i=15 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=14 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Catalan central binomial product at i=14 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=13 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Catalan central binomial product at i=13 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=12 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Catalan central binomial product at i=12 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=11 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Catalan central binomial product at i=11 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=10 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Catalan central binomial product at i=10 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=9 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Catalan central binomial product at i=9 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=8 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Catalan central binomial product at i=8 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=7 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Catalan central binomial product at i=7 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=6 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Catalan central binomial product at i=6 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=5 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Catalan central binomial product at i=5 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=4 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Catalan central binomial product at i=4 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=3 with exact rational arithmetic, not rounded decimals.
• S050 · premise_and_result_audit — Recompute Catalan central binomial product at i=3 with exact rational arithmetic, not rounded decimals.
• S049 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=2 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Recompute Catalan central binomial product at i=2 with exact rational arithmetic, not rounded decimals.
• S047 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=1 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Recompute Catalan central binomial product at i=1 with exact rational arithmetic, not rounded decimals.
• S045 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S044 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S043 · premise_and_result_audit — Recompute Catalan complete convolution for C_21 with exact rational arithmetic, not rounded decimals.
• S042 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S041 · premise_and_result_audit — Recompute Catalan complete convolution for C_20 with exact rational arithmetic, not rounded decimals.
• S040 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S039 · premise_and_result_audit — Recompute Catalan complete convolution for C_19 with exact rational arithmetic, not rounded decimals.
• S038 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S037 · premise_and_result_audit — Recompute Catalan complete convolution for C_18 with exact rational arithmetic, not rounded decimals.
• S036 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S035 · premise_and_result_audit — Recompute Catalan complete convolution for C_17 with exact rational arithmetic, not rounded decimals.
• S034 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S033 · premise_and_result_audit — Recompute Catalan complete convolution for C_16 with exact rational arithmetic, not rounded decimals.
• S032 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S031 · premise_and_result_audit — Recompute Catalan complete convolution for C_15 with exact rational arithmetic, not rounded decimals.
• S030 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S029 · premise_and_result_audit — Recompute Catalan complete convolution for C_14 with exact rational arithmetic, not rounded decimals.
• 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

(21, 538257874440, 22, 0, 24466267020, 24466267020, Yes)

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.