A353227 Expansion of e.g.f. (1 - x^3)^(-x).

1, 0, 0, 0, 24, 0, 0, 2520, 20160, 0, 1209600, 19958400, 79833600, 1556755200, 39956716800, 326918592000, 5056340889600, 148203095040000, 1867358997504000, 30411275102208000, 946128558735360000, 15965919428659200000, 293266062902292480000

Offset

0,5

Links

Table of n, a(n) for n=0..22.

Formula

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)!.

a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(k,n-3*k)|/k!.

a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / (3*exp(n)). - Vaclav Kotesovec, May 04 2022

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x^3)^(-x)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*log(1-x^3))))
(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;
(PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(k, n-3*k, 1))/k!);

Crossrefs

Cf. A066166, A351156, A353223, A353226.

Sequence in context: A202184 A368027 A357967 * A075406 A075404 A356304

Adjacent sequences: A353224 A353225 A353226 * A353228 A353229 A353230

Keyword

nonn

Author

Seiichi Manyama, May 01 2022

Status

approved