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Expansion of e.g.f exp(log(1+x)+2*log(1+x)^2).
1

%I #25 Oct 30 2022 08:59:21

%S 1,1,4,0,44,-220,2056,-19544,213216,-2571552,34036224,-489916416,

%T 7614555648,-127028029440,2262903109632,-42857715985920,

%U 859647858427392,-18200106158320128,405498290896693248,-9482120962982547456,232156555727228971008

%N Expansion of e.g.f exp(log(1+x)+2*log(1+x)^2).

%H Robert Israel, <a href="/A189424/b189424.txt">Table of n, a(n) for n = 0..444</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) = Sum_{m=1..n} (Sum_{k=m..n} (k!*binomial(m,k-m)*2^(k-m)*Stirling1(n,k))/m!), n > 0, a(0) = 1.

%p S:= series(exp(log(1+x)+2*log(1+x)^2), x, 31):

%p seq(coeff(S,x,j)*j!,j=0..30); # _Robert Israel_, Jan 30 2018

%t With[{nn=20},CoefficientList[Series[Exp[Log[1+x]+2Log[1+x]^2],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Dec 27 2014 *)

%o (Maxima)

%o a(n):=sum(sum(k!*binomial(m, k-m)*2^(k-m)*stirling1(n, k), k, m, n)/m!, m, 1, n);

%K sign

%O 0,3

%A _Vladimir Kruchinin_, Apr 21 2011