A257634 a(n) = (A001163(n)/A001164(n))*3*(2*n)!^2/n!!.

3, 1, 3, -1390, -139895, 2064875400, 999912530925, -128585633463727440, -176876516433064573125, 109242473594498195269718400, 333170810414553853376721961875, -698025623281503752808511373154720000, -4073023833462008382211035330291042675375

Offset: 0

Author

Vladimir Reshetnikov, Nov 04 2015

Keywords

• sign

Comments

Coefficients in Stirling's asymptotic expansion of the Gamma function, normalized to integers using factor 3*(2*n)!^2/n!!.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..30
• Eric Weisstein's World of Mathematics, Stirling's Series.

Crossrefs

Cf. A001163, A001164.

Programs

Maple

h := proc(k) option remember; local j; `if`(k=0,1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
g := n -> doublefactorial(2*n-1)*(2*n)!^2/doublefactorial(n):
seq(3*h(2*n)*g(n), n=0..12); # Peter Luschny, Nov 05 2015

Mathematica

Table[3 (2n)!^2/n!! (6n+1)!!/4^n Sum[(-1)^m 2^k StirlingS2[2n+k+m, m]/((2n+2k+1) (2n+k+m)! (2n-k)! (k-m)!), {k, 0, 2n}, {m, 0, k}], {n, 0, 12}]

Formula

a(n) = 3*(2*n)!*(6*n+1)!!/(n!!*4^n) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i)*C(i,j)*C(j,k) / ((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients.