A368650 a(n) = (6*n + 1)!*(9*n + 1)!/((2*n)!*(3*n)!*((5*n + 1)!)^2).

1, 2940, 27511848, 324265486545, 4234842288963000, 58626067532977225512, 842744763083824037236800, 12437726604034570811549435040, 187171833825593326056635733697560, 2859197188199406875783449346275416000, 44198453917285616202092687086145825181264, 689863061309915307698539343386922516078167200

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..11.

Formula

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).

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).

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.

Maple

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

Crossrefs

Cf. A304126, A368545, A082368, A113424.

Sequence in context: A365899 A236196 A054832 * A068290 A231313 A259999

Adjacent sequences: A368647 A368648 A368649 * A368651 A368652 A368653

Keyword

nonn

Author

Karol A. Penson, Jan 02 2024

Status

approved