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

%I A009306 #67 Mar 27 2022 03:18:24
%S A009306 0,1,1,-1,-2,9,6,-155,232,3969,-20870,-118779,1655028,1610257,
%T A009306 -143697722,522358005,13332842416,-138189937791,-1128293525646,
%U A009306 29219838555781,17274118159180,-5993074252801839,38541972209299966,1179892974640047669
%N A009306 Expansion of e.g.f.: log(1 + exp(x)*x).
%H A009306 Alois P. Heinz, <a href="/A009306/b009306.txt">Table of n, a(n) for n = 0..200</a>
%H A009306 Gottfried Helms, <a href="https://math.stackexchange.com/questions/4141754">Infinite sum of powerseries likely converges to a powerseries with rational coefficients ...</a> May 14, 2021
%H A009306 Vaclav Kotesovec, <a href="/A009306/a009306.jpg">Plot of (abs(a(n))/n!)^(1/n) for n = 1..1000</a>
%H A009306 Vladimir Kruchinin and D. V. Kruchinin, <a href="http://arxiv.org/abs/1103.2582">Composita and their properties</a>, arXiv:1103.2582 [math.CO], 2011-2013.
%F A009306 a(n) = n! * Sum_{k=1..n} k^(n-k-1) * (-1)^(k+1)/(n-k)!. - _Vladimir Kruchinin_, Sep 07 2010
%F A009306 a(n) = n - Sum_{k=1..n-1} binomial(n-1,k-1) * (n-k) * a(k). - _Ilya Gutkovskiy_, Jan 17 2020
%F A009306 Lim sup_{n->infinity} (abs(a(n))/n!)^(1/n) = 1/abs(LambertW(-1)) = 1/A238274. - _Vaclav Kotesovec_, May 26 2021
%p A009306 a:= n-> n! *add(k^(n-k-1) *(-1)^(k+1) /(n-k)!, k=1..n):
%p A009306 seq(a(n), n=0..25);
%t A009306 With[{nn=30},CoefficientList[Series[Log[1+Exp[x]x],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Oct 22 2016 *)
%o A009306 (PARI) seq(n)=Vec(serlaplace(log(1 + exp(x + O(x^n))*x)), -(n+1)) \\ _Andrew Howroyd_, May 26 2021
%Y A009306 Cf. A009444.
%K A009306 sign,easy
%O A009306 0,5
%A A009306 _R. H. Hardin_
%E A009306 Extended with signs by _Olivier Gérard_, Mar 15 1997
%E A009306 Definition clarified by _Harvey P. Dale_, Oct 22 2016