A353228 Expansion of e.g.f. (1 - x)^(-x^2).

1, 0, 0, 6, 12, 40, 540, 3528, 25200, 263520, 2741760, 30048480, 372794400, 4971957120, 70612686144, 1076056027200, 17469796780800, 300562292459520, 5468568356666880, 104917700221125120, 2116572758902425600, 44794683422986936320, 992435268252158438400

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..449

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2) * a(n-k)/(n-k)!.

a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(n-2*k,k)|/(n-2*k)!.

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

Mathematica

nmax = 20; CoefficientList[Series[(1-x)^(-x^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 12 2022 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^2)))
(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^2*log(1-x))))
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)*v[i-j+1]/(i-j)!)); v;
(PARI) a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Crossrefs

Cf. A066166, A351503, A351492, A353223, A353229.

Sequence in context: A052747 A007121 A356910 * A366752 A351503 A371118

Adjacent sequences: A353225 A353226 A353227 * A353229 A353230 A353231

Keyword

nonn

Author

Seiichi Manyama, May 01 2022

Status

approved