A368692 a(n) = (12*n + 6)!*(6*n + 9)!/(108*(4*n + 2)!*(2*n + 3)!*((6*n + 5)!)^2).

14, 563108, 54231252075, 6700034035890000, 928978310614152999200, 137569863175651804211692560, 21253098849879053645154605945160, 3381375421559384124434964404229384000, 549714622911935710495977183989400234273000

Offset

0,1

Comments

According to A. Adolphson and S. Sperber, "On the integrality of hypergeometric series whose coefficients are factorial ratios", ArXiv: 2001.03296, s.page 14, first equation after Eq.(7.4): for any two integers K, L, the ratios (3*K)!*(3*L)!/(K!*L!*((K+L)!)^2) are proven to be integers. 108*a(n) results from K = 4*n+2 and L = 2*n+3, n>=0. It is conjectured here that a(n) are integers.

Links

Table of n, a(n) for n=0..8.

A. Adolphson and S. Sperber, On the integrality of hypergeometric series whose coefficients are factorial ratios, Acta Arithmetica 200 (2021), no.1, 39-59.

Formula

G.f.: 14*hypergeometric8F7([7/12, 2/3, 5/6, 11/12, 13/12, 17/12, 13/6, 7/3], [1, 7/6, 4/3, 3/2, 3/2, 5/3, 11/6], 186624*z).

E.g.f.: 14*hypergeometric8F8([7/12, 2/3, 5/6, 11/12, 13/12, 17/12, 13/6, 7/3], [1, 1, 7/6, 4/3, 3/2, 3/2, 5/3, 11/6], 186624*z).

a(n) = Integral_{x=0..186624} x^n*W(x) dx, n>=0, where W(x) = (1/(20736*Pi))*MeijerG([[], [0, 0, 1/6, 1/3, 1/2, 1/2, 2/3, 5/6]], [[-5/12, -1/3, -1/6, -1/12, 1/12, 5/12, 7/6, 4/3], []], x/186624). MeijerG is the Meijer G - function. W(x) can be represented as an expression containing the sum of 4 generalized hypergeometric functions of type 8F7. W(x) is a positive function in the interval [0, 186624], is singular at x=0 and monotonically decreases to zero at x = 186624. This integral representation as the n-th power moment of the positive function W(x) in the interval [0, 186624] is unique, as W(x) is the solution of the Hausdorff moment problem.

Let b(n) = Gamma(7+ 12*n)/(6*Gamma(2 + 2*n)*Gamma(3 + 4*n)*Gamma(6 + 6*n)), then a(n) = b(n) * A272399(n+2). - Peter Luschny, Jan 06 2024

Maple

seq((12*n + 6)!*(6*n + 9)!/(108*(4*n + 2)!*(2*n + 3)!*((6*n + 5)!)^2), n=0..9);

Crossrefs

Cf. A368650, A304126, A368545, A082368, A113424, A368875.

Sequence in context: A372084 A159430 A013800 * A350891 A347626 A013870

Adjacent sequences: A368689 A368690 A368691 * A368693 A368694 A368695

Keyword

nonn

Author

Karol A. Penson, Jan 03 2024

Status

approved