A009306 Expansion of e.g.f.: log(1 + exp(x)*x).

0, 1, 1, -1, -2, 9, 6, -155, 232, 3969, -20870, -118779, 1655028, 1610257, -143697722, 522358005, 13332842416, -138189937791, -1128293525646, 29219838555781, 17274118159180, -5993074252801839, 38541972209299966, 1179892974640047669

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Gottfried Helms, Infinite sum of powerseries likely converges to a powerseries with rational coefficients ... May 14, 2021
• Vaclav Kotesovec, Plot of (abs(a(n))/n!)^(1/n) for n = 1..1000
• Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Crossrefs

Cf. A009444.

Programs

Maple

a:= n-> n! *add(k^(n-k-1) *(-1)^(k+1) /(n-k)!, k=1..n):
seq(a(n), n=0..25);

Mathematica

With[{nn=30},CoefficientList[Series[Log[1+Exp[x]x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 22 2016 *)

PARI

seq(n)=Vec(serlaplace(log(1 + exp(x + O(x^n))*x)), -(n+1)) \\ Andrew Howroyd, May 26 2021

Formula

a(n) = n! * Sum_{k=1..n} k^(n-k-1) * (-1)^(k+1)/(n-k)!. - Vladimir Kruchinin, Sep 07 2010

a(n) = n - Sum_{k=1..n-1} binomial(n-1,k-1) * (n-k) * a(k). - Ilya Gutkovskiy, Jan 17 2020

Lim sup_{n->infinity} (abs(a(n))/n!)^(1/n) = 1/abs(LambertW(-1)) = 1/A238274. - Vaclav Kotesovec, May 26 2021

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified by Harvey P. Dale, Oct 22 2016