A295433 a(n) = (12*n)!*n!/((8*n)!*(3*n)!*(2*n)!).

1, 990, 3432198, 13768454700, 58626071754822, 257898242928604740, 1158088115444301759900, 5274648294443835666191640, 24274428407219984638641697350, 112600884656246368465947735787860, 525586819747900180429387794022656948, 2465690419601453784410118364323664257000

Offset: 0

Author

Gheorghe Coserea, Nov 23 2017

Keywords

• nonn

Crossrefs

Cf. A295431, A495432.

Programs

Mathematica

Array[(12 #)!*#!/((8 #)!*(3 #)!*(2 #)!) &, 12, 0] (* Michael De Vlieger, Nov 23 2017 *)
CoefficientList[ Series[ HypergeometricPFQ[{1/12, 1/6, 5/12, 7/12, 5/6, 11/12}, {1/8, 3/8, 1/2, 5/8, 7/8}, 19683 x], {x, 0, 10}], x] (* Robert G. Wilson v, Nov 23 2017 *)

Formula

G.f.: hypergeom([1/12, 1/6, 5/12, 7/12, 5/6, 11/12], [1/8, 3/8, 1/2, 5/8, 7/8], 19683/4*x).

a(n) ~ 3^(9*n) / (sqrt(Pi*n) * 2^(2*n + 3/2)). - Vaclav Kotesovec, Oct 06 2021

From Karol A. Penson, Jun 13 2025: (Start)

a(n) = Integral_{x=0..19683/4} x^n*W(x)*dx, where W(x) is the Meijer G-function which in the Maple notation is equal to (sqrt(2)/(19683*sqrt(Pi)))*MeijerG([[],[0,-1/2,-7/8,-5/8,-3/8,-1/8]],[[-11/12,-5/6,-7/12,-5/12,-1/6,-1/12],[]],4*x/19683).

W(x) can be represented in terms of 6 different generalized hypergeometric functions of type 6F5 of argument 4*x/19683. W(x) is a positive, U-shaped function on x=(0,19683/4), and is singular at x=0 and at x=19683/4. Thus a(n) is a positive definite sequence. This representation is unique as W(x) is the solution of the Hausdorff moment problem. (End)