A091496 -id:A091496 - cp's OEIS frontend

A276098 a(n) = (7n)!(3/2n)!/((7n/2)!(3n)!(2n)!).

1, 48, 6006, 860160, 130378950, 20392706048, 3254013513660, 526470648692736, 86047769258554950, 14173603389190963200, 2349023203055914140756, 391249767795614684282880, 65434374898388743460014620, 10981406991821583404677201920

Offset: 0

Peter Bala, Aug 22 2016

Let a > b be nonnegative integers. The ratio of factorials (2*a*n)!*(b*n)!/( (a*n)!*(2*b*n)!*((a - b)*n)! ) is known to be an integer for n >= 0 (see, for example, Bober, Theorem 1.1). We have the companion result: Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for n >= 0. This is the case a = 3, b = 1. Other cases include A091496 (a = 2, b = 0), A091527 (a = 1, b = 0), A262732 (a = 2, b = 1), A262733 (a = 3, b = 2) and A276099 (a = 4, b = 2).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

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., 79, Issue 2, (2009), 422-444.

Cf. A046521, A091496, A091527, A262732, A262733, A276099.

seq(simplify((7*n)!*(3/2*n)!/((7*n/2)!*(3*n)!*(2*n)!)), n = 0..20);

from math import factorial
from sympy import factorial2
def A276098(n): return int((factorial(7*n)*factorial2(3*n)<<(n<<1))//factorial2(7*n)//factorial(3*n)//factorial(n<<1)) # Chai Wah Wu, Aug 10 2023

a(n) = Sum_{k = 0..2*n} binomial(7*n, 2*n - k)*binomial(3*n + k - 1, k).
a(n) = Sum_{k = 0..n} binomial(10*n, 2*n - 2*k)*binomial(3*n + k - 1, k).
Recurrence: a(n) = 28*(7*n - 1)*(7*n - 3)*(7*n - 5)*(7*n - 9)*(7*n - 11)*(7*n - 13)/(3*n*(n - 1)*(2*n - 1)*(2*n - 3)*(3*n - 1)*(3*n - 5)) * a(n-2).
a(n) ~ 1/sqrt(4*Pi*n) * (7^7/3^3)^(n/2).
O.g.f. A(x) = Hypergeom([13/14, 11/14, 9/14, 5/14, 3/14, 1/14], [5/6, 3/4, 1/2, 1/4, 1/6], (7^7/3^3)*x^2) + 48*x*Hypergeom([10/7, 9/7, 8/7, 6/7, 5/7, 4/7], [5/4, 4/3, 3/2, 3/4, 2/3], (7^7/3^3)*x^2).
a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^7/(1 - x)^3.
It follows that the o.g.f. A(x) equals the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*sqrt((1 - x)^3/(1 + x)^7) ) and put G(x) = 1 + x*d/dx(log(F(x))). Then A(x^2) = (G(x) + G(-x))/2.

A061162 a(n) = (6n)!n!/((3n)!(2n)!^2).

Original entry on oeis.org

1, 30, 2310, 204204, 19122246, 1848483780, 182327718300, 18236779032600, 1842826521244230, 187679234340049620, 19232182592635611060, 1980665038436368775400, 204826599735691440534300, 21255328931341321610645544, 2212241139727064219063537016

Offset: 0

Richard Stanley, Apr 17 2001

According to page 781 of the cited reference the generating function F(x) for a(n) is algebraic but not obviously so and the minimal polynomial satisfied by F(x) is quite large.

This sequence is the particular case a = 3, b = 1 of the following result (see Bober, Theorem 1.2): let a, b be nonnegative integers with a > b and gcd(a,b) = 1. Then (2*a*n)!*(b*n)!/((a*n)!*(2*b*n)!*((a-b)*n)!) is an integer for all integer n >= 0. Other cases include A211419 (a = 3, b = 2), A211420 (a = 4, b = 1) and A211421 (a = 4, b = 3) and A061163 (a = 5, b = 1). The o.g.f. Sum_{n >= 1} a(n)*z^n is algebraic over the field of rational functions Q(z) (see Rodriguez-Villegas). - Peter Bala, Apr 10 2012

M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited - 2001 and Beyond, Springer, Berlin, 2001, pp. 771-808.
R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

Harry J. Smith, Table of n, a(n) for n = 0..100
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, 2007, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., Vol. 79, Issue 2, (2009) 422-444.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22, p. 11.
W. Mlotkowski and Karol A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.
F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math/0701362 [math.NT], 2007.

Cf. A061163, A061164, A007297, A091527, A262732.

See also A091527, A211419, A211420, A211421.

A061162 := n->(6*n)!*n!/((3*n)!*(2*n)!^2);

a[n_] := 16^n Gamma[3 n + 1/2]/(Gamma[n + 1/2] Gamma[2 n + 1]);
Table[a[n], {n, 0, 14}] (* Peter Luschny, Mar 01 2018 *)

{ for (n=0, 100, write("b061162.txt", n, " ", (6*n)!*n!/((3*n)!*(2*n)!^2)) ) } \\ Harry J. Smith, Jul 18 2009

a(n) ~ 1/2*Pi^(-1/2)*n^(-1/2)*2^(2*n)*3^(3*n)*{1 - 1/72*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Jun 11 2002
n*(2*n-1)*a(n) -6*(6*n-1)*(6*n-5)*a(n-1)=0. - R. J. Mathar, Oct 26 2014
From Peter Bala, Aug 21 2016: (Start)
a(n) = Sum_{k = 0..2*n} binomial(6*n, k)*binomial(4*n - k - 1, 2*n - k).
a(n) = Sum_{k = 0..n} binomial(8*n, 2*n - 2*k)*binomial(2*n + k - 1, k).
O.g.f. A(x) = Hypergeom([5/6, 1/6], [1/2], 108*x).
a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^6/(1 - x)^2. Cf. A091496 and A262732. It follows that the o.g.f. A(x) for this sequence is the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.
Let G(x) = 1/x * series reversion( x*(1 - x)/(1 + x)^3 ) = 1 + 4*x + 23*x^2 + 156*x^3 + 1162*x^4 + ..., essentially the o.g.f. for A007297. Then A(x^2) equals the even part of 1 + x*(d/dx log(G(x))).
exp(Sum_{n >= 1} a(n)*x^n/n) = F(x), where F(x) = 1 + 30*x + 1605*x^2 + 107218*x^3 + 8043114*x^4 + 647773116*x^5 + 54730094637*x^6 + ... has integer coefficients since F(x^2) = G(x)*G(-x). Furthermore, F(x)^(1/6) = 1 + 5*x + 205*x^2 + 12328*x^3 + 874444*x^4 + 68022261*x^5 + 5613007167*x^6 + ... appears to have all integer coefficients. (End)
a(n) is the n-th moment of the positive weight function w(x) on x = (0,108), i.e.: a(n) = Integral_{x=0..108} x^n*w(x) dx, n >= 0, where w(x) = sqrt(3)*(1 + sqrt(1 - x/108))^(2/3)/(12*2^(1/3)*Pi*x^(5/6)*sqrt(1 - x/108)) + 2^(4/3)*sqrt(3)/(864*Pi*x^(1/6)*(1 + sqrt(1 - x/108))^(2/3)*sqrt(1 - x/108)). The weight function w(x) is singular at x=0 and at x=108 and is the solution of the Hausdorff moment problem. This solution is unique. - Karol A. Penson, Mar 01 2018
a(n) = 2^(4*n)*binomial(-n-1/2, 2*n). - Ira M. Gessel, Jan 04 2025

A211417 Integral factorial ratio sequence: a(n) = (30n)!n!/((15n)!(10n)!(6*n)!).

Original entry on oeis.org

1, 77636318760, 53837289804317953893960, 43880754270176401422739454033276880, 38113558705192522309151157825210540422513019720, 34255316578084325260482016910137568877961925210286281393760

Offset: 0

Peter Bala, Apr 11 2012

The integrality of this sequence can be used to prove Chebyshev's estimate C(1)*x/log(x) <= #{primes <= x} <= C(2)*x/log(x), for x sufficiently large; the constant C(1) = 0.921292... and C(2) = 1.105550.... Chebyshev's approach used the related step function floor(x) -floor(x/2) -floor(x/3) -floor(x/5) +floor(x/30). See A182067.
This sequence is one of the 52 sporadic integral factorial ratio sequences of height 1 found by V. I. Vasyunin.
The o.g.f. sum {n >= 0} a(n)*z^n is a generalized hypergeometric series of type 8F7 (see Bober, Table 2, Entry 31) and is an algebraic function of degree 483840 over the field of rational functions Q(z) (see Rodriguez-Villegas). Bober remarks that the monodromy group of the differential equation satisfied by the o.g.f. is W(E_8), the Weyl group of the E_8 root system.
See the Bala link for the proof that a(n), n = 0,1,2..., is an integer.
Congruences: a(p^k) == a(p^(k-1)) ( mod p^(3*k) ) for any prime p >= 5 and any positive integer k (write a(n) as C(30*n,15*n)*C(15*n,5*n)/C(6*n,n) and use equation 39 in Mestrovic, p. 12). More generally, the congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) may hold for any prime p >= 5 and any positive integers n and k. Cf. A295431. - Peter Bala, Jan 24 2020

N. J. A. Sloane, Table of n, a(n) for n = 0..50
Peter Bala, Proof of the integrality of A211417 and A211418
Frits Beukers, Hypergeometric functions, how special are they?, Notices Amer. Math. Soc. 61 (2014), no. 1, 48--56. MR3137256
J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, arXiv:0709.1977 [math.NT], 2007; J. London Math. Soc., 79, Issue 2, (2009), 422-444.
Florian Fürnsinn and Sergey Yurkevich, Algebraicity of hypergeometric functions with arbitrary parameters, arXiv:2308.12855 [math.CA], 2023.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Fernando Rodriguez Villegas, Integral ratios of factorials and algebraic hypergeometric functions, arXiv:math.NT/0701362, 2007.
Fernando Rodriguez Villegas, Mixed Hodge numbers and factorial ratios, arXiv:1907.02722 [math.NT], 2019.
K. Soundararajan, Integral Factorial Ratios, arXiv:1901.05133 [math.NT], 2019.
Wadim Zudilin, Integer-valued factorial ratios, MathOverflow question 26336, 2010.

Cf. A182067, A211418, A061162, A061163, A061164, A091496, A091527, A112292, A182400, A211419, A211420, A211421, A276100, A262733, A295431.

[Factorial(30*n)*Factorial(n)/(Factorial(15*n)*Factorial(10*n)*Factorial(6*n)): n in [0..10]]; // Vincenzo Librandi, Oct 03 2015

Table[(30 n)!*n!/((15 n)!*(10 n)!*(6 n)!), {n, 0, 5}] (* Michael De Vlieger, Oct 02 2015 *)

a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!);
vector(10, n, a(n-1)) \\ Altug Alkan, Oct 02 2015

a(n) ~ 2^(14*n-1) * 3^(9*n-1/2) * 5^(5*n-1/2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 30 2016

A276099 a(n) = (9n)!(5/2n)!/((9n/2)!(5n)!(2n)!).

Original entry on oeis.org

1, 96, 24310, 7028736, 2149374150, 678057476096, 218191487357116, 71184392021606400, 23459604526110889542, 7791432263086689484800, 2603575153867220801823060, 874329826463740757819785216, 294822072977645830504963830300

Offset: 0

Peter Bala, Aug 22 2016

Let a > b be nonnegative integers. The ratio of factorials (2*a*n)!*(b*n)!/( (a*n)!*(2*b*n)!*((a - b)*n)! ) is known to be an integer for all integer n >= 0 (see, for example, Bober, Theorem 1.1). We have the companion result: Let a > b be nonnegative integers. Then the ratio of factorials ((2*a + 1)*n)!*((b + 1/2)*n)!/(((a + 1/2)*n)!*((2*b + 1)*n)!*((a - b)*n)!) is an integer for all integer n >= 0. This is the case a = 4, b = 2. Other cases include A091496 (a = 2, b = 0), A091527 (a = 1, b = 0), A262732 (a = 2, b = 1), A262733 (a = 3, b = 2) and A276098 (a = 3, b = 1).

R. P. Stanley, Enumerative Combinatorics Volume 2, Cambridge Univ. Press, 1999, Theorem 6.33, p. 197.

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., 79, Issue 2, (2009), 422-444.

Cf. A091496, A091527, A262732, A262733, A276098.

seq(simplify((9*n)!*(5/2*n)!/((9*n/2)!*(5*n)!*(2*n)!)), n = 0..20);

Table[((9n)!(5/2 n)!)/((9 n/2)!(5n)!(2n)!),{n,0,15}] (* Harvey P. Dale, May 21 2024 *)

from math import factorial
from sympy import factorial2
def A276099(n): return int((factorial(9*n)*factorial2(5*n)<<(n<<1))//factorial2(9*n)//factorial(5*n)//factorial(n<<1)) # Chai Wah Wu, Aug 10 2023

a(n) = Sum_{k = 0..2*n} binomial(9*n, k)*binomial(7*n - k - 1, 2*n - k).
a(n) = Sum_{k = 0..n} binomial(14*n, 2*n - 2*k)*binomial(5*n + k - 1, k).
a(n) ~ 1/sqrt(4*Pi*n) * (3^18/5^5)^(n/2).
O.g.f. A(x) = Hypergeom([17/18, 13/18, 11/18, 7/18, 5/18, 1/18, 5/6, 1/6], [9/10, 7/10, 3/10, 1/10, 3/4, 1/4, 1/2], (3^18/5^5)*x^2) + 96*x*Hypergeom([13/9, 11/9, 10/9, 8/9, 7/9, 5/9, 4/3, 2/3], [7/5, 6/5, 4/5, 3/5, 5/4, 3/4, 3/2], (3^18/5^5)*x^2).
a(n) = [x^(2*n)] H(x)^n, where H(x) = (1 + x)^9/(1 - x)^5.
It follows that the o.g.f. A(x) for this sequence is the diagonal of the bivariate rational generating function 1/2*( 1/(1 - t*H(sqrt(x))) + 1/(1 - t*H(-sqrt(x))) ) and hence is algebraic by Stanley 1999, Theorem 6.33, p. 197.
Let F(x) = (1/x)*Series_Reversion( x*sqrt((1 - x)^5/(1 + x)^9) ) and put G(x) = 1 + x*d/dx(log(F(x))). Then A(x^2) = (G(x) + G(-x))/2.

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).

cp's OEIS Frontend

A276098 a(n) = (7*n)!*(3/2*n)!/((7*n/2)!*(3*n)!*(2*n)!).

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A061162 a(n) = (6n)!n!/((3n)!(2n)!^2).

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A211417 Integral factorial ratio sequence: a(n) = (30*n)!*n!/((15*n)!*(10*n)!*(6*n)!).

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A276099 a(n) = (9*n)!*(5/2*n)!/((9*n/2)!*(5*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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A276098 a(n) = (7n)!(3/2n)!/((7n/2)!(3n)!(2n)!).

A211417 Integral factorial ratio sequence: a(n) = (30n)!n!/((15n)!(10n)!(6*n)!).

A276099 a(n) = (9n)!(5/2n)!/((9n/2)!(5n)!(2n)!).