A368650 - OEIS

%I #17 Jan 02 2024 09:36:02

%S 1,2940,27511848,324265486545,4234842288963000,58626067532977225512,

%T 842744763083824037236800,12437726604034570811549435040,

%U 187171833825593326056635733697560,2859197188199406875783449346275416000,44198453917285616202092687086145825181264,689863061309915307698539343386922516078167200

%N a(n) = (6*n + 1)!*(9*n + 1)!/((2*n)!*(3*n)!*((5*n + 1)!)^2).

%C a(n) can be rigorously proven to be an integer for n>=0.

%F G.f.: hypergeometric10F9([2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 5/6, 8/9, 10/9, 7/6], [2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1, 6/5, 6/5], (167365651248*z)/9765625).

%F O.g.f.: hypergeometric10F10([2/9, 1/3, 4/9, 5/9, 2/3, 7/9, 5/6, 8/9, 10/9, 7/6], [2/5, 2/5, 3/5, 3/5, 4/5, 4/5, 1, 1, 6/5, 6/5], (167365651248*z)/9765625).

%F a(n) = Integral_{x=0..167365651248/9765625} x^n*W(x) dx, n>=0, where W(x) = (78125*MeijerG([[], [-3/5, -3/5, -2/5, -2/5, -1/5, -1/5, 0, 0, 1/5, 1/5]], [[1/6, 1/9, -1/9, -1/6, -2/9, -1/3, -4/9, -5/9, -2/3, -7/9], []], (9765625*x)/167365651248))/(2066242608*Pi). MeijerG is the Meijer G - function. W(x) can be represented as a sum of 10 hypergeometric functions of type 10F9. W(x) can be proven to be a positive function in the interval [0, 167365651248/9765625]. W(x) is singular at x=0 and monotonically decreases to zero at x = 167365651248/9765625. This integral representation as the n-th power moment of the positive function  W(x) in the interval [0, 167365651248/9765625] is unique, as W(x) is the solution of the Hausdorff moment problem.

%p seq((6*n + 1)!*(9*n + 1)!/((2*n)!*(3*n)!*((5*n + 1)!)^2), n=0..12);

%Y Cf. A304126, A368545, A082368, A113424.

%K nonn

%O 0,2

%A _Karol A. Penson_, Jan 02 2024