A295431 -id:A295431 - cp's OEIS frontend

A295471 a(n) = (20n)!n!/((10n)!(8n)!(3*n)!).

1, 2771340, 44524428808860, 835982760936614190900, 16702642470437308383606668060, 345116202503279265243707597937393840, 7282855378270617096441349556073829143864900, 155927544469969434644890704895476696960247603510200

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

Table[((20n)!n!)/((10n)!(8n)!(3n)!),{n,0,10}] (* Harvey P. Dale, Feb 03 2021 *)

G.f.: hypergeom([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/8, 1/3, 3/8, 1/2, 5/8, 2/3, 7/8], 625000000/27*x).

A295475 a(n) = (20n)!(3n)!/((10n)!(9n)!(4n)!).

Original entry on oeis.org

1, 461890, 935387159850, 2182519096151533552, 5398397695681095146608490, 13787702861800799166026014363140, 35936545440404705429404600374145350960, 94988519542968717273112119046397042704228800, 253668511451664737174164792111754605640400674756970

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

Table[((20n)!(3n)!)/((10n)!(9n)!(4n)!),{n,0,20}] (* Harvey P. Dale, Mar 26 2022 *)

G.f.: hypergeom([1/20, 3/20, 7/20, 9/20, 11/20, 13/20, 17/20, 19/20], [1/9, 2/9, 4/9, 1/2, 5/9, 7/9, 8/9], 40960000000000/14348907*x).

A295477 a(n) = (24n)!n!/((12n)!(8n)!(5*n)!).

Original entry on oeis.org

1, 267711444, 527048385075849780, 1217325447549161369383451760, 2994861478939539397101967737771147060, 7622711741038504461830231565042108084502603944, 19819090822775008829253654773280241192988848189242236400, 52286737777519489251960880501419064028764839670925596855165438440

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/24, 5/24, 7/24, 11/24, 13/24, 17/24, 19/24, 23/24], [1/5, 1/4, 2/5, 1/2, 3/5, 3/4, 4/5], 8916100448256/3125*x).

A295478 a(n) = (24n)!(5n)!(2n)!/((12n)!(10n)!(8n)!*n!).

Original entry on oeis.org

1, 2124694, 17116035801030, 156955248839610299260, 1520823957776376583242698310, 15195903357109656115403419318791444, 154846359560326628948765847970927516623900, 1599554665194478490546369090543508578392036048440

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..142

Cf. A295431.

Table[((24n)!(5n)!(2n)!)/((12n)!(10n)!(8n)!n!),{n,0,10}] (* Harvey P. Dale, Jun 14 2025 *)

G.f.: hypergeom([1/24, 5/24, 7/24, 11/24, 13/24, 17/24, 19/24, 23/24], [1/10, 1/4, 3/10, 1/2, 7/10, 3/4, 9/10], 34828517376/3125*x).

A295479 a(n) = (24n)!(4n)!n!/((12n)!(8n)!(7n)!(2*n)!).

Original entry on oeis.org

1, 76488984, 36856530424884600, 20728451893251973782071160, 12401082728528113445556802226795640, 7671567513095586883562392061857092727662984, 4846567811298033077517533116990723918586593960411800, 3106360084464723603791702457062194851072644408605122994989000

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/24, 5/24, 7/24, 11/24, 13/24, 17/24, 19/24, 23/24], [1/7, 2/7, 3/7, 1/2, 4/7, 5/7, 6/7], 570630428688384/823543*x).

A295481 a(n) = (24n)!(4n)!(3n)!/((12n)!(9n)!(8n)!(2n)!).

Original entry on oeis.org

1, 6374082, 180669266788650, 5882199787281395215344, 202857467914154836183288657770, 7221430962039777689508936047385667332, 262302851034768406033478290202841394160147760, 9661245477414039526585354113472313766826824134193600

Offset: 0

Gheorghe Coserea, Nov 28 2017

nonn

Cf. A295431.

G.f.: hypergeom([1/24, 5/24, 7/24, 11/24, 13/24, 17/24, 19/24, 23/24], [1/9, 2/9, 4/9, 1/2, 5/9, 7/9, 8/9], 1073741824/27*x).

A364518 Square array read by ascending antidiagonals: T(n,k) = [x^(2*k)] ( (1 + x)^(n+2)/(1 - x)^(n-2) )^k for n, k >= 0.

Original entry on oeis.org

1, 1, -2, 1, 0, 6, 1, 6, -10, -20, 1, 16, 70, 0, 70, 1, 30, 630, 924, 198, -252, 1, 48, 2310, 28672, 12870, 0, 924, 1, 70, 6006, 204204, 1385670, 184756, -4420, -3432, 1, 96, 12870, 860160, 19122246, 69206016, 2704156, 0, 12870, 1, 126, 24310, 2704156, 130378950, 1848483780, 3528923580, 40116600, 104006, -48620

Offset: 0

Peter Bala, Aug 07 2023

Compare with A364303 and A364519.
Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L), where c_1 + ... + c_K = d_1 + ... + d_L, we can define the factorial ratio sequence u_n(c, d) = (c_1*n)!*(c_2*n)!* ... *(c_K*n)!/ ( (d_1*n)!*(d_2*n)!* ... *(d_L*n)! ) and ask whether it is integral for all n >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1 (see A295431). Soundararajan gives many examples of two-parameter families of integral factorial ratio sequences of height 2.
Each row of the present table is an integral factorial ratio sequence of height 1. It is usually assumed that the c's and d's are integers but here some of the c's and d's are half-integers. See A276098 and the cross references there for further examples of this type.
It is known that the unsigned version of row 0 (the central binomial numbers A000984) and row 2 satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that all the row sequences of the table satisfy the same supercongruences.

			 Square array begins:
 n\k|  0   1      2        3           4             5
  - + - - - - - - - - - - - - - - - - - - - - - - - - -
  0 |  1  -2      6      -20          70          -252   ... see A000984
  1 |  1   0    -10        0         198             0   ... see A211419
  2 |  1   6     70      924       12870        184756   ... A001448
  3 |  1  16    630    28672     1385670      69206016   ... A091496
  4 |  1  30   2310   204204    19122246    1848483780   ... A061162
  5 |  1  48   6006   860160   130378950   20392706048   ... A276098
  6 |  1  70  12870  2704156   601080390  137846528820   ... A001448 bisected
  7 |  1  96  24310  7028736  2149374150  678057476096   ... A276099

Peter Bala, Some integer ratios of factorials
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
K. Soundararajan, Integral factorial ratios: irreducible examples with height larger than 1, Phil. Trans. Royal Soc., A378: 2018044, 2019.
Wikipedia, Dixon's identity
Wikipedia, Hypergeometric function

Cf. A000984 (row 0 unsigned), A211419 (row 1 unsigned without 0's), A001448 (row 2), A091496 (row 3), A061162 (row 4), A276098 (row 5), A001448 bisected (row 6), A276099 (row 7).

Cf. A330843, A364303, A364506, A364509, A364513, A364519.

T(n,k) = add( binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j), j = 0..2*k):
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);

T(n,k) = sum(j = 0, 2*k, binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j));
lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ Michel Marcus, Aug 13 2023

T(n,k) = Sum_{j = 0..2*k} binomial((n+2)*k, j)*binomial(n*k-j-1, 2*k-j).
T(2,k) = binomial(4*k,2*k).
For n >= 3, T(n,k) = binomial(n*k-1,2*k) * hypergeom([-(n+2)*k, -2*k], [1 - n*k], -1) except when (n,k) = (3,1).
For n >= 2, T(n,k) = ((n+2)*k)!*((n-2)*k/2)!/(((n+2)*k/2)!*((n-2)*k)!*(2*k)!) by Kummer's Theorem.
T(n,k) = [x^k] (1 - x)^(2*k) * Chebyshev_T(n*k, (1 + x)/(1 - x)).
T(n,k) = Sum_{j = 0..k} binomial(2*n*k, 2*j)*binomial((n-1)*k-j-1, k-j).
For n >= 3, T(n,k) = binomial((n-1)*k-1,k) * hypergeom([-n*k, -k, -n*k + 1/2], [1 - (n-1)*k, 1/2], 1).
The row generating functions are algebraic functions over the field of rational functions Q(x).

A364519 Square array read by ascending antidiagonals: T(n,k) = [x^(3*k)] ( (1 + x)^(n+3)/(1 - x)^(n-3) )^k for n, k >= 0.

Original entry on oeis.org

1, 1, 0, 1, -4, -20, 1, 0, 28, 0, 1, 20, -84, -220, 924, 1, 64, 924, 0, 1820, 0, 1, 140, 12012, 48620, 16796, -15504, -48620, 1, 256, 60060, 2621440, 2704156, 0, 134596, 0, 1, 420, 204204, 29745716, 608435100, 155117520, -3801900, -1184040, 2704156, 1, 640, 554268, 187432960, 15628090140, 146028888064, 9075135300, 0, 10518300, 0

Offset: 0

Peter Bala, Aug 07 2023

Compare with A364303 and A364518.
Given two sequences of integers c = (c_1, c_2, ..., c_K) and d = (d_1, d_2, ..., d_L), where c_1 + ... + c_K = d_1 + ... + d_L, we can define the factorial ratio sequence u_n(c, d) = (c_1*n)!*(c_2*n)!* ... *(c_K*n)!/ ( (d_1*n)!*(d_2*n)!* ... *(d_L*n)! ) and ask whether it is integral for all n >= 0. The integer L - K is called the height of the sequence. Bober completed the classification of integral factorial ratio sequences of height 1 (see A295431).
Each row of the present table is an integral factorial ratio sequence of height 1. It is usually assumed that the c's and d's are integers but here some of the c's and d's are half-integers. See A276098 and the cross references there for further examples of this type.
It is known that A005810, the unsigned version of row 1, satisfies the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5 and all positive integers n and r. We conjecture that each row sequence of the table satisfies the same supercongruences.

			Square array begins:
 n\k| 0    1       2          3             4                5
  - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  0 | 1    0     -20          0           924                0  ... see A066802
  1 | 1   -4      28       -220          1820           -15504  ... see A005810
  2 | 1    0     -84          0         16796                0
  3 | 1   20     924      48620       2704156        155117520  ... A066802
  4 | 1   64   12012    2621440     608435100     146028888064  ... A364520
  5 | 1  140   60060   29745716   15628090140    8480843582640  ... A211420

Peter Bala, Some integer ratios of factorials
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc. (2) 79 2009, 422-444.
Wikipedia, Hypergeometric function

Cf. A066802 (row 3, also row 0 unsigned and without 0's), A005810 (row 1 unsigned), A364520 (row 4), A211420 (row 5).

Cf. A330843, A364303, A364518.

T(n,k) := add( binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j), j = 0..3*k):
# display as a square array
seq(print(seq(T(n, k), k = 0..10)), n = 0..10);
# display as a sequence
seq(seq(T(n-k, k), k = 0..n), n = 0..10);

T(n,k) = sum(j = 0, 3*k, binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j));
lista(nn) = for( n=0, nn, for (k=0, n, print1(T(n-k, k), ", "))); \\ Michel Marcus, Aug 13 2023

T(n,k) = Sum_{j = 0..3*k} binomial((n+3)*k, j)*binomial(n*k-j-1, 3*k-j).
For n >= 3, T(n,k) = binomial(n*k-1,3*k) * hypergeom([-(n+3)*k, -3*k], [1 - n*k], -1) = ((n+3)*k)!*((n-3)*k/2)!/(((n+3)*k/2)!*((n-3)*k)!*(3*k)!) by Kummer's Theorem.
The row generating functions are algebraic functions over the field of rational functions Q(x).

A295434 a(n) = (12n)!(3n)!/((8n)!(6n)!*n!).

Original entry on oeis.org

1, 99, 22287, 5663700, 1517599215, 418974963099, 117912668329092, 33631821857441592, 9687910353619450095, 2811928784155292169225, 821085376403131135219287, 240930004359452293067067600, 70981901511015157572343133700, 20983636110818804086505029904100, 6221192124050298514903984159997400

Offset: 0

Gheorghe Coserea, Nov 23 2017

nonn

Cf. A295431.

Array[(12 #)!*(3 #)!/((8 #)!*(6 #)!*#!) &, 15, 0] (* Michael De Vlieger, Nov 23 2017 *)
CoefficientList[ Series[ HypergeometricPFQ[{1/12, 1/3, 5/12, 7/12, 2/3, 11/12}, {1/8, 3/8, 1/2, 5/8, 7/8}, 19683/64 x], {x, 0, 14}], x] (* Robert G. Wilson v, Nov 23 2017 *)

G.f.: hypergeom([1/12, 1/3, 5/12, 7/12, 2/3, 11/12], [1/8, 3/8, 1/2, 5/8, 7/8], 19683/64*x).

a(n) ~ 3^(9*n + 1/2) / (2^(6*n + 3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 11 2025

A295436 a(n) = (12n)!(5n)!/((10n)!(4n)!(3n)!).

Original entry on oeis.org

1, 110, 31878, 10550540, 3693446214, 1333894333860, 491399166995100, 183538361769733080, 69248533171932254790, 26330347618877971057844, 10073025233727857639656628, 3872722823191369534267881000, 1495040784249187268901385878300, 579143282269456580741926257778920

Offset: 0

Gheorghe Coserea, Nov 23 2017

nonn

Cf. A295431.

Array[(12#)!(5#)!/((10#)!(4#)!(3#)!) &, 14, 0] (* Michael De Vlieger, Nov 23 2017 *)
CoefficientList[ Series[ HypergeometricPFQ[{1/12, 1/6, 5/12, 7/12, 5/6, 11/12}, {1/10, 3/10, 1/2, 7/10, 9/10}, 1259712/3125 x], {x, 0, 13}], x] (* Robert G. Wilson v, Nov 23 2017 *)

G.f.: hypergeom([1/12, 1/6, 5/12, 7/12, 5/6, 11/12], [1/10, 3/10, 1/2, 7/10, 9/10], 1259712/3125*x).

a(n) ~ 2^(6*n-1) * 3^(9*n) / (5^(5*n) * sqrt(Pi*n)). - Vaclav Kotesovec, Jul 11 2025

cp's OEIS Frontend

A295471 a(n) = (20*n)!*n!/((10*n)!*(8*n)!*(3*n)!).

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A295475 a(n) = (20*n)!*(3*n)!/((10*n)!*(9*n)!*(4*n)!).

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A295477 a(n) = (24*n)!*n!/((12*n)!*(8*n)!*(5*n)!).

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A295478 a(n) = (24*n)!*(5*n)!*(2*n)!/((12*n)!*(10*n)!*(8*n)!*n!).

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A295479 a(n) = (24*n)!*(4*n)!*n!/((12*n)!*(8*n)!*(7*n)!*(2*n)!).

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A295481 a(n) = (24*n)!*(4*n)!*(3*n)!/((12*n)!*(9*n)!*(8*n)!*(2*n)!).

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A364518 Square array read by ascending antidiagonals: T(n,k) = [x^(2*k)] ( (1 + x)^(n+2)/(1 - x)^(n-2) )^k for n, k >= 0.

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Maple

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A364519 Square array read by ascending antidiagonals: T(n,k) = [x^(3*k)] ( (1 + x)^(n+3)/(1 - x)^(n-3) )^k for n, k >= 0.

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Maple

PARI

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A295434 a(n) = (12*n)!*(3*n)!/((8*n)!*(6*n)!*n!).

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Mathematica

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A295436 a(n) = (12*n)!*(5*n)!/((10*n)!*(4*n)!*(3*n)!).

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A295471 a(n) = (20n)!n!/((10n)!(8n)!(3*n)!).

A295475 a(n) = (20n)!(3n)!/((10n)!(9n)!(4n)!).

A295477 a(n) = (24n)!n!/((12n)!(8n)!(5*n)!).

A295478 a(n) = (24n)!(5n)!(2n)!/((12n)!(10n)!(8n)!*n!).

A295479 a(n) = (24n)!(4n)!n!/((12n)!(8n)!(7n)!(2*n)!).

A295481 a(n) = (24n)!(4n)!(3n)!/((12n)!(9n)!(8n)!(2n)!).

A295434 a(n) = (12n)!(3n)!/((8n)!(6n)!*n!).

A295436 a(n) = (12n)!(5n)!/((10n)!(4n)!(3n)!).