|
%I #21 May 04 2022 05:11:10
%S 1,0,0,0,24,0,0,2520,20160,0,1209600,19958400,79833600,1556755200,
%T 39956716800,326918592000,5056340889600,148203095040000,
%U 1867358997504000,30411275102208000,946128558735360000,15965919428659200000,293266062902292480000
%N Expansion of e.g.f. (1 - x^3)^(-x).
%F a(0) = 1; a(n) = (n-1)! * Sum_{k=2..floor((n+2)/3)} (3*k-2)/(k-1) * a(n-3*k+2)/(n-3*k+2)!.
%F a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/k!.
%F a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - _Vaclav Kotesovec_, May 04 2022
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(-x)))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3))))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=2, (i+2)\3, (3*j-2)/(j-1)*v[i-3*j+3]/(i-3*j+2)!)); v;
%o (PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(k, n-3*k, 1))/k!);
%Y Cf. A066166, A351156, A353223, A353226.
%K nonn
%O 0,5
%A _Seiichi Manyama_, May 01 2022
|
