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Expansion of e.g.f.: log(1 + exp(x)*x).
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%I #67 Mar 27 2022 03:18:24

%S 0,1,1,-1,-2,9,6,-155,232,3969,-20870,-118779,1655028,1610257,

%T -143697722,522358005,13332842416,-138189937791,-1128293525646,

%U 29219838555781,17274118159180,-5993074252801839,38541972209299966,1179892974640047669

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

%H Alois P. Heinz, <a href="/A009306/b009306.txt">Table of n, a(n) for n = 0..200</a>

%H 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 Vaclav Kotesovec, <a href="/A009306/a009306.jpg">Plot of (abs(a(n))/n!)^(1/n) for n = 1..1000</a>

%H 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 a(n) = n! * Sum_{k=1..n} k^(n-k-1) * (-1)^(k+1)/(n-k)!. - _Vladimir Kruchinin_, Sep 07 2010

%F a(n) = n - Sum_{k=1..n-1} binomial(n-1,k-1) * (n-k) * a(k). - _Ilya Gutkovskiy_, Jan 17 2020

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

%p a:= n-> n! *add(k^(n-k-1) *(-1)^(k+1) /(n-k)!, k=1..n):

%p seq(a(n), n=0..25);

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

%o (PARI) seq(n)=Vec(serlaplace(log(1 + exp(x + O(x^n))*x)), -(n+1)) \\ _Andrew Howroyd_, May 26 2021

%Y Cf. A009444.

%K sign,easy

%O 0,5

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997

%E Definition clarified by _Harvey P. Dale_, Oct 22 2016