This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A257634 #16 Feb 16 2025 08:33:25
%S A257634 3,1,3,-1390,-139895,2064875400,999912530925,-128585633463727440,
%T A257634 -176876516433064573125,109242473594498195269718400,
%U A257634 333170810414553853376721961875,-698025623281503752808511373154720000,-4073023833462008382211035330291042675375
%N A257634 a(n) = (A001163(n)/A001164(n))*3*(2*n)!^2/n!!.
%C A257634 Coefficients in Stirling's asymptotic expansion of the Gamma function, normalized to integers using factor 3*(2*n)!^2/n!!.
%H A257634 G. C. Greubel, <a href="/A257634/b257634.txt">Table of n, a(n) for n = 0..30</a>
%H A257634 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingsSeries.html">Stirling's Series.</a>
%F A257634 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.
%p A257634 h := proc(k) option remember; local j; `if`(k=0,1,
%p A257634 (h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
%p A257634 g := n -> doublefactorial(2*n-1)*(2*n)!^2/doublefactorial(n):
%p A257634 seq(3*h(2*n)*g(n), n=0..12); # _Peter Luschny_, Nov 05 2015
%t A257634 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}]
%Y A257634 Cf. A001163, A001164.
%K A257634 sign
%O A257634 0,1
%A A257634 _Vladimir Reshetnikov_, Nov 04 2015