%I #20 Mar 07 2020 12:09:12
%S 1,3,12,135,288,2835,51840,8505,2488320,12629925,209018880,492567075,
%T 75246796800,1477701225,902961561600,39565450299375,86684309913600,
%U 2255230667064375,514904800886784000,6765692001193125,86504006548979712000,7002491221234884375
%N Denominators of rational coefficients related to Stirling's asymptotic series for the Gamma function.
%C See A264148 for definitions and cross-references.
%H G. C. Greubel, <a href="/A264149/b264149.txt">Table of n, a(n) for n = 0..589</a>
%p h := proc(k) option remember; local j; `if`(k<=0, 1,
%p (h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
%p SGGS := n -> h(n)*doublefactorial(n-1):
%p A264149 := n -> denom(SGGS(n));
%p seq(A264149(n), n=0..26);
%t h[k_]:= h[k] = If[k <= 0, 1, (h[k - 1]/k - Sum[h[k - j]*h[j]/(j + 1), {j, 1, k - 1}])/(1 + 1/(k + 1))]; a[n_]:= h[n]*Factorial2[n - 1] // Denominator; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 09 2018 *)
%o (Sage)
%o def A264149(n):
%o @cached_function
%o def h(k):
%o if k<=0: return 1
%o S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
%o return (h(k-1)/k-S)/(1+1/(k+1))
%o return denominator(h(n)*(n-1).multifactorial(2))
%o print([A264149(n) for n in (0..21)])
%Y Cf. numerators in A264148.
%K nonn,frac
%O 0,2
%A _Peter Luschny_, Nov 05 2015