This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A368692 #27 Jan 10 2024 08:00:24
%S A368692 14,563108,54231252075,6700034035890000,928978310614152999200,
%T A368692 137569863175651804211692560,21253098849879053645154605945160,
%U A368692 3381375421559384124434964404229384000,549714622911935710495977183989400234273000
%N A368692 a(n) = (12*n + 6)!*(6*n + 9)!/(108*(4*n + 2)!*(2*n + 3)!*((6*n + 5)!)^2).
%C A368692 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.
%H A368692 A. Adolphson and S. Sperber, <a href="https://doi.org/10.4064/aa200427-5-4">On the integrality of hypergeometric series whose coefficients are factorial ratios</a>, Acta Arithmetica 200 (2021), no.1, 39-59.
%F A368692 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).
%F A368692 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).
%F A368692 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.
%F A368692 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
%p A368692 seq((12*n + 6)!*(6*n + 9)!/(108*(4*n + 2)!*(2*n + 3)!*((6*n + 5)!)^2),n=0..9);
%Y A368692 Cf. A368650, A304126, A368545, A082368, A113424, A368875.
%K A368692 nonn
%O A368692 0,1
%A A368692 _Karol A. Penson_, Jan 03 2024