Question 22775: The Recurrence and Both Closed Forms Use the Same Zero-Based Index

Original question and givens

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

• G001 — Use natural index n=24≥0, C_0=1 and C_(n+1)=sum over i=0..n of C_i*C_(n−i). Exact BigInt coefficients and answers are retained. Do not shift to a C_1-based convention or truncate a nonintegral quotient. Give (n, recurrence_value, closed_value, difference_value, all_equal, certificate).

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-recurrence-closed-cross-check;original snapshot and mathematics retained

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

ORIGINAL requested natural index n=24

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 complete convolution for C_22 := sum(24466267020, 6564120420, 3534526380, 2388193500, 1815027060, 1485022140, 1279719540, 1147334760, 1062347000, 1011354344, 987369656, 987369656, 1011354344, 1062347000, 1147334760, 1279719540, 1485022140, 1815027060, 2388193500, 3534526380, 6564120420, 24466267020)

Catalan complete convolution for C_22 = 91482563640

ORIGINAL index j=21;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 24466267020, 24466267020], [1, 1, 6564120420, 6564120420], [2, 2, 1767263190, 3534526380], [3, 5, 477638700, 2388193500], [4, 14, 129644790, 1815027060], [5, 42, 35357670, 1485022140], [6, 132, 9694845, 1279719540], [7, 429, 2674440, 1147334760], [8, 1430, 742900, 1062347000], [9, 4862, 208012, 1011354344], [10, 16796, 58786, 987369656], [11, 58786, 16796, 987369656], [12, 208012, 4862, 1011354344], [13, 742900, 1430, 1062347000], [14, 2674440, 429, 1147334760], [15, 9694845, 132, 1279719540], [16, 35357670, 42, 1485022140], [17, 129644790, 14, 1815027060], [18, 477638700, 5, 2388193500], [19, 1767263190, 2, 3534526380], [20, 6564120420, 1, 6564120420], [21, 24466267020, 1, 24466267020]]

C_22=91482563640;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, 24466267020]

Catalan complete convolution for C_23 := sum(91482563640, 24466267020, 13128240840, 8836315950, 6686941800, 5445081180, 4667212440, 4159088505, 3824449200, 3611979800, 3493769552, 3455793796, 3493769552, 3611979800, 3824449200, 4159088505, 4667212440, 5445081180, 6686941800, 8836315950, 13128240840, 24466267020, 91482563640)

Catalan complete convolution for C_23 = 343059613650

ORIGINAL index j=22;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 91482563640, 91482563640], [1, 1, 24466267020, 24466267020], [2, 2, 6564120420, 13128240840], [3, 5, 1767263190, 8836315950], [4, 14, 477638700, 6686941800], [5, 42, 129644790, 5445081180], [6, 132, 35357670, 4667212440], [7, 429, 9694845, 4159088505], [8, 1430, 2674440, 3824449200], [9, 4862, 742900, 3611979800], [10, 16796, 208012, 3493769552], [11, 58786, 58786, 3455793796], [12, 208012, 16796, 3493769552], [13, 742900, 4862, 3611979800], [14, 2674440, 1430, 3824449200], [15, 9694845, 429, 4159088505], [16, 35357670, 132, 4667212440], [17, 129644790, 42, 5445081180], [18, 477638700, 14, 6686941800], [19, 1767263190, 5, 8836315950], [20, 6564120420, 2, 13128240840], [21, 24466267020, 1, 24466267020], [22, 91482563640, 1, 91482563640]]

C_23=343059613650;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, 24466267020, 91482563640]

Catalan complete convolution for C_24 := sum(343059613650, 91482563640, 48932534040, 32820602100, 24741684660, 20060825400, 17113112280, 15168440430, 13863628350, 13003127280, 12477748400, 12228193432, 12228193432, 12477748400, 13003127280, 13863628350, 15168440430, 17113112280, 20060825400, 24741684660, 32820602100, 48932534040, 91482563640, 343059613650)

Catalan complete convolution for C_24 = 1289904147324

ORIGINAL index j=23;ALL [i,C_i,C_(j-i),exact product] rows=[[0, 1, 343059613650, 343059613650], [1, 1, 91482563640, 91482563640], [2, 2, 24466267020, 48932534040], [3, 5, 6564120420, 32820602100], [4, 14, 1767263190, 24741684660], [5, 42, 477638700, 20060825400], [6, 132, 129644790, 17113112280], [7, 429, 35357670, 15168440430], [8, 1430, 9694845, 13863628350], [9, 4862, 2674440, 13003127280], [10, 16796, 742900, 12477748400], [11, 58786, 208012, 12228193432], [12, 208012, 58786, 12228193432], [13, 742900, 16796, 12477748400], [14, 2674440, 4862, 13003127280], [15, 9694845, 1430, 13863628350], [16, 35357670, 429, 15168440430], [17, 129644790, 132, 17113112280], [18, 477638700, 42, 20060825400], [19, 1767263190, 14, 24741684660], [20, 6564120420, 5, 32820602100], [21, 24466267020, 2, 48932534040], [22, 91482563640, 1, 91482563640], [23, 343059613650, 1, 343059613650]]

C_24=1289904147324;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, 24466267020, 91482563640, 343059613650]

Catalan central:ORIGINAL binomial N=48,K=24

use k=min(K,N-K)=24;initialize exact product1;empty product for k0 is1

Catalan central binomial product at i=1 := mul(1, 25)

Catalan central binomial product at i=1 = 25

Catalan central exact binomial division by i=1 := div(25, 1)

Catalan central exact binomial division by i=1 = 25

Catalan central binomial product at i=2 := mul(25, 26)

Catalan central binomial product at i=2 = 650

Catalan central exact binomial division by i=2 := div(650, 2)

Catalan central exact binomial division by i=2 = 325

Catalan central binomial product at i=3 := mul(325, 27)

Catalan central binomial product at i=3 = 8775

Catalan central exact binomial division by i=3 := div(8775, 3)

Catalan central exact binomial division by i=3 = 2925

Catalan central binomial product at i=4 := mul(2925, 28)

Catalan central binomial product at i=4 = 81900

Catalan central exact binomial division by i=4 := div(81900, 4)

Catalan central exact binomial division by i=4 = 20475

Catalan central binomial product at i=5 := mul(20475, 29)

Catalan central binomial product at i=5 = 593775

Catalan central exact binomial division by i=5 := div(593775, 5)

Catalan central exact binomial division by i=5 = 118755

Catalan central binomial product at i=6 := mul(118755, 30)

Catalan central binomial product at i=6 = 3562650

Catalan central exact binomial division by i=6 := div(3562650, 6)

Catalan central exact binomial division by i=6 = 593775

Catalan central binomial product at i=7 := mul(593775, 31)

Catalan central binomial product at i=7 = 18407025

Catalan central exact binomial division by i=7 := div(18407025, 7)

Catalan central exact binomial division by i=7 = 2629575

Catalan central binomial product at i=8 := mul(2629575, 32)

Catalan central binomial product at i=8 = 84146400

Catalan central exact binomial division by i=8 := div(84146400, 8)

Catalan central exact binomial division by i=8 = 10518300

Catalan central binomial product at i=9 := mul(10518300, 33)

Catalan central binomial product at i=9 = 347103900

Catalan central exact binomial division by i=9 := div(347103900, 9)

Catalan central exact binomial division by i=9 = 38567100

Catalan central binomial product at i=10 := mul(38567100, 34)

Catalan central binomial product at i=10 = 1311281400

Catalan central exact binomial division by i=10 := div(1311281400, 10)

Catalan central exact binomial division by i=10 = 131128140

Catalan central binomial product at i=11 := mul(131128140, 35)

Catalan central binomial product at i=11 = 4589484900

Catalan central exact binomial division by i=11 := div(4589484900, 11)

Catalan central exact binomial division by i=11 = 417225900

Catalan central binomial product at i=12 := mul(417225900, 36)

Catalan central binomial product at i=12 = 15020132400

Catalan central exact binomial division by i=12 := div(15020132400, 12)

Catalan central exact binomial division by i=12 = 1251677700

Catalan central binomial product at i=13 := mul(1251677700, 37)

Catalan central binomial product at i=13 = 46312074900

Catalan central exact binomial division by i=13 := div(46312074900, 13)

Catalan central exact binomial division by i=13 = 3562467300

Catalan central binomial product at i=14 := mul(3562467300, 38)

Catalan central binomial product at i=14 = 135373757400

Catalan central exact binomial division by i=14 := div(135373757400, 14)

Catalan central exact binomial division by i=14 = 9669554100

Catalan central binomial product at i=15 := mul(9669554100, 39)

Catalan central binomial product at i=15 = 377112609900

Catalan central exact binomial division by i=15 := div(377112609900, 15)

Catalan central exact binomial division by i=15 = 25140840660

Catalan central binomial product at i=16 := mul(25140840660, 40)

Catalan central binomial product at i=16 = 1005633626400

Catalan central exact binomial division by i=16 := div(1005633626400, 16)

Catalan central exact binomial division by i=16 = 62852101650

Catalan central binomial product at i=17 := mul(62852101650, 41)

Catalan central binomial product at i=17 = 2576936167650

Catalan central exact binomial division by i=17 := div(2576936167650, 17)

Catalan central exact binomial division by i=17 = 151584480450

Catalan central binomial product at i=18 := mul(151584480450, 42)

Catalan central binomial product at i=18 = 6366548178900

Catalan central exact binomial division by i=18 := div(6366548178900, 18)

Catalan central exact binomial division by i=18 = 353697121050

Catalan central binomial product at i=19 := mul(353697121050, 43)

Catalan central binomial product at i=19 = 15208976205150

Catalan central exact binomial division by i=19 := div(15208976205150, 19)

Catalan central exact binomial division by i=19 = 800472431850

Catalan central binomial product at i=20 := mul(800472431850, 44)

Catalan central binomial product at i=20 = 35220787001400

Catalan central exact binomial division by i=20 := div(35220787001400, 20)

Catalan central exact binomial division by i=20 = 1761039350070

Catalan central binomial product at i=21 := mul(1761039350070, 45)

Catalan central binomial product at i=21 = 79246770753150

Catalan central exact binomial division by i=21 := div(79246770753150, 21)

Catalan central exact binomial division by i=21 = 3773655750150

Catalan central binomial product at i=22 := mul(3773655750150, 46)

Catalan central binomial product at i=22 = 173588164506900

Catalan central exact binomial division by i=22 := div(173588164506900, 22)

Catalan central exact binomial division by i=22 = 7890371113950

Catalan central binomial product at i=23 := mul(7890371113950, 47)

Catalan central binomial product at i=23 = 370847442355650

Catalan central exact binomial division by i=23 := div(370847442355650, 23)

Catalan central exact binomial division by i=23 = 16123801841550

Catalan central binomial product at i=24 := mul(16123801841550, 48)

Catalan central binomial product at i=24 = 773942488394400

Catalan central exact binomial division by i=24 := div(773942488394400, 24)

Catalan central exact binomial division by i=24 = 32247603683100

Catalan ORIGINAL denominator n plus1 := sum(24, 1)

Catalan ORIGINAL denominator n plus1 = 25

ORIGINAL n=24;central binomial=32247603683100;positive denominator n+1=25

Euclidean quotient=1289904147324,remainder=0;check remainder0 BEFORE treating quotient as Catalan integer

Catalan exact closed quotient after remainder0 := div(32247603683100, 25)

Catalan exact closed quotient after remainder0 = 1289904147324

Catalan adjacent:ORIGINAL binomial N=48,K=25

use k=min(K,N-K)=23;initialize exact product1;empty product for k0 is1

Catalan adjacent binomial product at i=1 := mul(1, 26)

Catalan adjacent binomial product at i=1 = 26

Catalan adjacent exact binomial division by i=1 := div(26, 1)

Catalan adjacent exact binomial division by i=1 = 26

Catalan adjacent binomial product at i=2 := mul(26, 27)

Catalan adjacent binomial product at i=2 = 702

Catalan adjacent exact binomial division by i=2 := div(702, 2)

Catalan adjacent exact binomial division by i=2 = 351

Catalan adjacent binomial product at i=3 := mul(351, 28)

Catalan adjacent binomial product at i=3 = 9828

Catalan adjacent exact binomial division by i=3 := div(9828, 3)

Catalan adjacent exact binomial division by i=3 = 3276

Catalan adjacent binomial product at i=4 := mul(3276, 29)

Catalan adjacent binomial product at i=4 = 95004

Catalan adjacent exact binomial division by i=4 := div(95004, 4)

Catalan adjacent exact binomial division by i=4 = 23751

Catalan adjacent binomial product at i=5 := mul(23751, 30)

Catalan adjacent binomial product at i=5 = 712530

Catalan adjacent exact binomial division by i=5 := div(712530, 5)

Catalan adjacent exact binomial division by i=5 = 142506

Catalan adjacent binomial product at i=6 := mul(142506, 31)

Catalan adjacent binomial product at i=6 = 4417686

Catalan adjacent exact binomial division by i=6 := div(4417686, 6)

Catalan adjacent exact binomial division by i=6 = 736281

Catalan adjacent binomial product at i=7 := mul(736281, 32)

Catalan adjacent binomial product at i=7 = 23560992

Catalan adjacent exact binomial division by i=7 := div(23560992, 7)

Catalan adjacent exact binomial division by i=7 = 3365856

Catalan adjacent binomial product at i=8 := mul(3365856, 33)

Catalan adjacent binomial product at i=8 = 111073248

Catalan adjacent exact binomial division by i=8 := div(111073248, 8)

Catalan adjacent exact binomial division by i=8 = 13884156

Catalan adjacent binomial product at i=9 := mul(13884156, 34)

Catalan adjacent binomial product at i=9 = 472061304

Catalan adjacent exact binomial division by i=9 := div(472061304, 9)

Catalan adjacent exact binomial division by i=9 = 52451256

Catalan adjacent binomial product at i=10 := mul(52451256, 35)

Catalan adjacent binomial product at i=10 = 1835793960

Catalan adjacent exact binomial division by i=10 := div(1835793960, 10)

Catalan adjacent exact binomial division by i=10 = 183579396

Catalan adjacent binomial product at i=11 := mul(183579396, 36)

Catalan adjacent binomial product at i=11 = 6608858256

Catalan adjacent exact binomial division by i=11 := div(6608858256, 11)

Catalan adjacent exact binomial division by i=11 = 600805296

Catalan adjacent binomial product at i=12 := mul(600805296, 37)

Catalan adjacent binomial product at i=12 = 22229795952

Catalan adjacent exact binomial division by i=12 := div(22229795952, 12)

Catalan adjacent exact binomial division by i=12 = 1852482996

Catalan adjacent binomial product at i=13 := mul(1852482996, 38)

Catalan adjacent binomial product at i=13 = 70394353848

Catalan adjacent exact binomial division by i=13 := div(70394353848, 13)

Catalan adjacent exact binomial division by i=13 = 5414950296

Catalan adjacent binomial product at i=14 := mul(5414950296, 39)

Catalan adjacent binomial product at i=14 = 211183061544

Catalan adjacent exact binomial division by i=14 := div(211183061544, 14)

Catalan adjacent exact binomial division by i=14 = 15084504396

Catalan adjacent binomial product at i=15 := mul(15084504396, 40)

Catalan adjacent binomial product at i=15 = 603380175840

Catalan adjacent exact binomial division by i=15 := div(603380175840, 15)

Catalan adjacent exact binomial division by i=15 = 40225345056

Catalan adjacent binomial product at i=16 := mul(40225345056, 41)

Catalan adjacent binomial product at i=16 = 1649239147296

Catalan adjacent exact binomial division by i=16 := div(1649239147296, 16)

Catalan adjacent exact binomial division by i=16 = 103077446706

Catalan adjacent binomial product at i=17 := mul(103077446706, 42)

Catalan adjacent binomial product at i=17 = 4329252761652

Catalan adjacent exact binomial division by i=17 := div(4329252761652, 17)

Catalan adjacent exact binomial division by i=17 = 254661927156

Catalan adjacent binomial product at i=18 := mul(254661927156, 43)

Catalan adjacent binomial product at i=18 = 10950462867708

Catalan adjacent exact binomial division by i=18 := div(10950462867708, 18)

Catalan adjacent exact binomial division by i=18 = 608359048206

Catalan adjacent binomial product at i=19 := mul(608359048206, 44)

Catalan adjacent binomial product at i=19 = 26767798121064

Catalan adjacent exact binomial division by i=19 := div(26767798121064, 19)

Catalan adjacent exact binomial division by i=19 = 1408831480056

Catalan adjacent binomial product at i=20 := mul(1408831480056, 45)

Catalan adjacent binomial product at i=20 = 63397416602520

Catalan adjacent exact binomial division by i=20 := div(63397416602520, 20)

Catalan adjacent exact binomial division by i=20 = 3169870830126

Catalan adjacent binomial product at i=21 := mul(3169870830126, 46)

Catalan adjacent binomial product at i=21 = 145814058185796

Catalan adjacent exact binomial division by i=21 := div(145814058185796, 21)

Catalan adjacent exact binomial division by i=21 = 6943526580276

Catalan adjacent binomial product at i=22 := mul(6943526580276, 47)

Catalan adjacent binomial product at i=22 = 326345749272972

Catalan adjacent exact binomial division by i=22 := div(326345749272972, 22)

Catalan adjacent exact binomial division by i=22 = 14833897694226

Catalan adjacent binomial product at i=23 := mul(14833897694226, 48)

Catalan adjacent binomial product at i=23 = 712027089322848

Catalan adjacent exact binomial division by i=23 := div(712027089322848, 23)

Catalan adjacent exact binomial division by i=23 = 30957699535776

Catalan central minus adjacent binomial := sub(32247603683100, 30957699535776)

Catalan central minus adjacent binomial = 1289904147324

ORIGINAL n=24;recurrence value=1289904147324;central=32247603683100,adjacent=30957699535776,denominator=25

closed quotient=1289904147324;binomial difference=1289904147324;ALL THREE agree;balanced paths counted by choose(2n,n),bad paths first-crossing reflection counted by choose(2n,n+1)

Reverse checks

• S151 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S150 · premise_and_result_audit — Recompute Catalan central minus adjacent binomial with exact rational arithmetic, not rounded decimals.
• S149 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=23 with exact rational arithmetic, not rounded decimals.
• S148 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=23 with exact rational arithmetic, not rounded decimals.
• S147 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=22 with exact rational arithmetic, not rounded decimals.
• S146 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=22 with exact rational arithmetic, not rounded decimals.
• S145 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=21 with exact rational arithmetic, not rounded decimals.
• S144 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=21 with exact rational arithmetic, not rounded decimals.
• S143 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=20 with exact rational arithmetic, not rounded decimals.
• S142 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=20 with exact rational arithmetic, not rounded decimals.
• S141 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=19 with exact rational arithmetic, not rounded decimals.
• S140 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=19 with exact rational arithmetic, not rounded decimals.
• S139 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=18 with exact rational arithmetic, not rounded decimals.
• S138 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=18 with exact rational arithmetic, not rounded decimals.
• S137 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=17 with exact rational arithmetic, not rounded decimals.
• S136 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=17 with exact rational arithmetic, not rounded decimals.
• S135 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=16 with exact rational arithmetic, not rounded decimals.
• S134 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=16 with exact rational arithmetic, not rounded decimals.
• S133 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=15 with exact rational arithmetic, not rounded decimals.
• S132 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=15 with exact rational arithmetic, not rounded decimals.
• S131 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=14 with exact rational arithmetic, not rounded decimals.
• S130 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=14 with exact rational arithmetic, not rounded decimals.
• S129 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=13 with exact rational arithmetic, not rounded decimals.
• S128 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=13 with exact rational arithmetic, not rounded decimals.
• S127 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=12 with exact rational arithmetic, not rounded decimals.
• S126 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=12 with exact rational arithmetic, not rounded decimals.
• S125 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=11 with exact rational arithmetic, not rounded decimals.
• S124 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=11 with exact rational arithmetic, not rounded decimals.
• S123 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=10 with exact rational arithmetic, not rounded decimals.
• S122 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=10 with exact rational arithmetic, not rounded decimals.
• S121 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=9 with exact rational arithmetic, not rounded decimals.
• S120 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=9 with exact rational arithmetic, not rounded decimals.
• S119 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=8 with exact rational arithmetic, not rounded decimals.
• S118 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=8 with exact rational arithmetic, not rounded decimals.
• S117 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=7 with exact rational arithmetic, not rounded decimals.
• S116 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=7 with exact rational arithmetic, not rounded decimals.
• S115 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=6 with exact rational arithmetic, not rounded decimals.
• S114 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=6 with exact rational arithmetic, not rounded decimals.
• S113 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=5 with exact rational arithmetic, not rounded decimals.
• S112 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=5 with exact rational arithmetic, not rounded decimals.
• S111 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=4 with exact rational arithmetic, not rounded decimals.
• S110 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=4 with exact rational arithmetic, not rounded decimals.
• S109 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=3 with exact rational arithmetic, not rounded decimals.
• S108 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=3 with exact rational arithmetic, not rounded decimals.
• S107 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=2 with exact rational arithmetic, not rounded decimals.
• S106 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=2 with exact rational arithmetic, not rounded decimals.
• S105 · premise_and_result_audit — Recompute Catalan adjacent exact binomial division by i=1 with exact rational arithmetic, not rounded decimals.
• S104 · premise_and_result_audit — Recompute Catalan adjacent binomial product at i=1 with exact rational arithmetic, not rounded decimals.
• S103 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S102 · premise_and_result_audit — Recompute Catalan exact closed quotient after remainder0 with exact rational arithmetic, not rounded decimals.
• S101 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S100 · premise_and_result_audit — Recompute Catalan ORIGINAL denominator n plus1 with exact rational arithmetic, not rounded decimals.
• S099 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=24 with exact rational arithmetic, not rounded decimals.
• S098 · premise_and_result_audit — Recompute Catalan central binomial product at i=24 with exact rational arithmetic, not rounded decimals.
• S097 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=23 with exact rational arithmetic, not rounded decimals.
• S096 · premise_and_result_audit — Recompute Catalan central binomial product at i=23 with exact rational arithmetic, not rounded decimals.
• S095 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=22 with exact rational arithmetic, not rounded decimals.
• S094 · premise_and_result_audit — Recompute Catalan central binomial product at i=22 with exact rational arithmetic, not rounded decimals.
• S093 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=21 with exact rational arithmetic, not rounded decimals.
• S092 · premise_and_result_audit — Recompute Catalan central binomial product at i=21 with exact rational arithmetic, not rounded decimals.
• S091 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=20 with exact rational arithmetic, not rounded decimals.
• S090 · premise_and_result_audit — Recompute Catalan central binomial product at i=20 with exact rational arithmetic, not rounded decimals.
• S089 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=19 with exact rational arithmetic, not rounded decimals.
• S088 · premise_and_result_audit — Recompute Catalan central binomial product at i=19 with exact rational arithmetic, not rounded decimals.
• S087 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=18 with exact rational arithmetic, not rounded decimals.
• S086 · premise_and_result_audit — Recompute Catalan central binomial product at i=18 with exact rational arithmetic, not rounded decimals.
• S085 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=17 with exact rational arithmetic, not rounded decimals.
• S084 · premise_and_result_audit — Recompute Catalan central binomial product at i=17 with exact rational arithmetic, not rounded decimals.
• S083 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=16 with exact rational arithmetic, not rounded decimals.
• S082 · premise_and_result_audit — Recompute Catalan central binomial product at i=16 with exact rational arithmetic, not rounded decimals.
• S081 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=15 with exact rational arithmetic, not rounded decimals.
• S080 · premise_and_result_audit — Recompute Catalan central binomial product at i=15 with exact rational arithmetic, not rounded decimals.
• S079 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=14 with exact rational arithmetic, not rounded decimals.
• S078 · premise_and_result_audit — Recompute Catalan central binomial product at i=14 with exact rational arithmetic, not rounded decimals.
• S077 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=13 with exact rational arithmetic, not rounded decimals.
• S076 · premise_and_result_audit — Recompute Catalan central binomial product at i=13 with exact rational arithmetic, not rounded decimals.
• S075 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=12 with exact rational arithmetic, not rounded decimals.
• S074 · premise_and_result_audit — Recompute Catalan central binomial product at i=12 with exact rational arithmetic, not rounded decimals.
• S073 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=11 with exact rational arithmetic, not rounded decimals.
• S072 · premise_and_result_audit — Recompute Catalan central binomial product at i=11 with exact rational arithmetic, not rounded decimals.
• S071 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=10 with exact rational arithmetic, not rounded decimals.
• S070 · premise_and_result_audit — Recompute Catalan central binomial product at i=10 with exact rational arithmetic, not rounded decimals.
• S069 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=9 with exact rational arithmetic, not rounded decimals.
• S068 · premise_and_result_audit — Recompute Catalan central binomial product at i=9 with exact rational arithmetic, not rounded decimals.
• S067 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=8 with exact rational arithmetic, not rounded decimals.
• S066 · premise_and_result_audit — Recompute Catalan central binomial product at i=8 with exact rational arithmetic, not rounded decimals.
• S065 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=7 with exact rational arithmetic, not rounded decimals.
• S064 · premise_and_result_audit — Recompute Catalan central binomial product at i=7 with exact rational arithmetic, not rounded decimals.
• S063 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=6 with exact rational arithmetic, not rounded decimals.
• S062 · premise_and_result_audit — Recompute Catalan central binomial product at i=6 with exact rational arithmetic, not rounded decimals.
• S061 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=5 with exact rational arithmetic, not rounded decimals.
• S060 · premise_and_result_audit — Recompute Catalan central binomial product at i=5 with exact rational arithmetic, not rounded decimals.
• S059 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=4 with exact rational arithmetic, not rounded decimals.
• S058 · premise_and_result_audit — Recompute Catalan central binomial product at i=4 with exact rational arithmetic, not rounded decimals.
• S057 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=3 with exact rational arithmetic, not rounded decimals.
• S056 · premise_and_result_audit — Recompute Catalan central binomial product at i=3 with exact rational arithmetic, not rounded decimals.
• S055 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=2 with exact rational arithmetic, not rounded decimals.
• S054 · premise_and_result_audit — Recompute Catalan central binomial product at i=2 with exact rational arithmetic, not rounded decimals.
• S053 · premise_and_result_audit — Recompute Catalan central exact binomial division by i=1 with exact rational arithmetic, not rounded decimals.
• S052 · premise_and_result_audit — Recompute Catalan central binomial product at i=1 with exact rational arithmetic, not rounded decimals.
• S051 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S050 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S049 · premise_and_result_audit — Recompute Catalan complete convolution for C_24 with exact rational arithmetic, not rounded decimals.
• S048 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S047 · premise_and_result_audit — Recompute Catalan complete convolution for C_23 with exact rational arithmetic, not rounded decimals.
• S046 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S045 · premise_and_result_audit — Recompute Catalan complete convolution for C_22 with exact rational arithmetic, not rounded decimals.
• 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

(24, 1289904147324, 1289904147324, 1289904147324, 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.