A368875 a(n) = 24*(3*n + 1)!/(n!*((n + 2)!)^2).

6, 16, 105, 1008, 12012, 164736, 2494206, 40646320, 701149020, 12655450080, 237026033790, 4577828250240, 90739095674400, 1838979005667840, 37993593597567210, 798259862714284080, 17022152442879594780, 367791659430639444000, 8040845154302354844450

Offset: 0

Author

Karol A. Penson, Jan 08 2024

Keywords

• nonn

Comments

According to A. Adolphson and S. Sperber (see Links), see page 14, second equation after Eq.(7.4): for any two integers K, L, the ratios (3*K+1)!*(3*L+1)!/(K!*L!*((K+L+1)!)^2) are proven to be integers. Here a(n) results from K = 1 and L = n, n >= 0.

Links

• A. Adolphson and S. Sperber, On the integrality of hypergeometric series whose coefficients are factorial ratios, arXiv:2001.03296 [math.NT], 2020.
• A. Adolphson and S. Sperber, On the integrality of hypergeometric series whose coefficients are factorial ratios, Acta Arithmetica 200 (2021), no.1, 39-59.

Crossrefs

Cf. A368650, A304126, A368545, A082368, A113424, A368692.

Programs

Maple

seq(24*(3*n + 1)!/(n!*((n + 2)!)^2),n=0..17);

Mathematica

Table[24*(3*n + 1)!/(n!*((n + 2)!)^2),{n,0,16}] (* James C. McMahon, Jan 08 2024 *)

SageMath

def a(n): return (24 * (n + 1) * (n + 2) * gamma(3*n + 2)) / gamma(n + 3)^3
print([a(n) for n in range(19)])  # Peter Luschny, Jan 09 2024

Formula

G.f.: 6*hypergeometric3F2([2/3, 1, 4/3], [3, 3], 27*z).

G.f.: -(hypergeometric2F1([-4/3, -2/3], [1], 27*z) - 1)/(3*z^2) + 8/z.

E.g.f.: 6*hypergeometric3F3([2/3, 1, 4/3], [3, 3, 1], 27*z).

a(n) = Integral_{x=0..27} x^n*W(x) dx, n >= 0, where

W(x) = (243*2^(2/3)*Gamma(5/6)*Gamma(2/3)*hypergeometric2F1([-4/3, -4/3], [1/3], x/27)) / (16*Pi^(5/2)*x^(1/3)) - (3*sqrt(3)*2^(1/3)*x^(1/3)* hypergeometric2F1([-2/3, -2/3], [5/3], x/27))/(2*sqrt(Pi)*Gamma(5/6)* Gamma(2/3)).

W(x) is a positive function in the interval [0, 27], is singular at x = 0 with the singularity x^(-1/3), and monotonically decreases to zero at x = 27, with W'(x) tending to zero at x = 27. This integral representation as the n-th power moment of the positive function W(x) in the interval [0, 27] is unique, as W(x) is the solution of the Hausdorff moment problem.