A375604 Expansion of e.g.f. 1 / (exp(-x^2) - x).

1, 1, 4, 18, 108, 840, 7680, 82320, 1009680, 13910400, 213071040, 3589850880, 65975152320, 1313624632320, 28166959941120, 647099547494400, 15857424488505600, 412878579034521600, 11382450106662835200, 331230511848421785600, 10146149192841050188800

Offset: 0

Author

Seiichi Manyama, Aug 21 2024

Keywords

• nonn

Crossrefs

Cf. A072597, A375607.

Cf. A375394, A375608.

Programs

Mathematica

With[{nn=20},CoefficientList[Series[1/(Exp[-x^2]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 01 2025 *)

PARI

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x^2)-x)))

PARI

a(n) = n!*sum(k=0, n\2, (n-2*k+1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n-2*k+1)^k/k!.

a(n) ~ sqrt(Pi) * 2^(n/2 + 1) * n^(n + 1/2) / ((1 + LambertW(2)) * exp(n) * LambertW(2)^((n+1)/2)). - Vaclav Kotesovec, Aug 21 2024