A368513 a(n) = (6*n+1)!/(n!*(2*n)!*(3*n+1)!).

1, 105, 25740, 7759752, 2574148500, 902522205585, 328074738591600, 122332313750680800, 46485667563689950596, 17924037162454524601500, 6991900927489809108938160, 2753354160571011216583946400, 1092796344333659321191117573200, 436609643814534385348768088729640, 175431288302508215774213129529432000

Offset: 0

Author

Karol A. Penson, Dec 28 2023

Keywords

• nonn

Crossrefs

Cf. A113424.

Programs

Maple

seq((6*n+1)!/(n!*(2*n)!*(3*n+1)!),n=0..14)

Formula

G.f.: hypergeometric3F2([1/3, 5/6, 7/6], [1, 4/3], 432*z).

E.g.f.: hypergeometric3F3([1/3, 5/6, 7/6], [1, 1, 4/3], 432*z).

a(n) = Integral_{x=0..432} x^n*W(x) dx, n>=0, where W(x) = sqrt(3)/(12*Pi*x^(2/3)) - gamma(2/3)*gamma(5/6)*sqrt(3)*hypergeometric3F2([1/2, 5/6, 5/6], [2/3, 3/2], x/432)/(432*Pi^(5/2)*x^(1/6)) + x^(1/6)*hypergeometric3F2([5/6, 7/6, 7/6], [4/3, 11/6], x/432)/(12960*sqrt(Pi)*gamma(2/3)*gamma(5/6)). W(x) is positive on x = [0, 432], it diverges at x=0, and monotonically decreases for x>0. It appears that at x=432, W(x) tends to a constant value close to 0.0007368284. This integral representation as the n-th power moment of the positive function W(x) on the interval [0, 432] is unique, as W(x) is the solution of the Hausdorff moment problem.

The shape of W(x) in the above integral representation of a(n) resembles very much the shape of the corresponding W(x) in A113424.