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%I #25 Mar 16 2024 11:19:31
%S 1,1,5,12,53,152,361,-168,-16055,-123200,-779827,-2504832,18694397,
%T 338660480,3543246193,19320001536,-64409565935,-2591458500608,
%U -36445173712747,-254934852857856,809555224961861,46263863308992512,744209131179612025,5617003290092961792
%N Expansion of e.g.f. exp(sin(x)+2*sin(x)^2).
%H Vincenzo Librandi, <a href="/A189421/b189421.txt">Table of n, a(n) for n = 0..65</a>
%H Vladimir Kruchinin, 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, (binomial(m,k-m)*((-1)^(n-k)+1)*sum(i=0..k/2, (2*i-k)^n*binomial(k,i)*(-1)^((n+k)/2-i))))/(2^m*m!)), n>0, a(0)=1.
%t a[n_] := Sum[Sum[Binomial[m, k-m] ((-1)^(n-k)+1) Sum[(2i-k)^n Binomial[k, i] (-1)^((n+k)/2-i), {i, 0, k/2}], {k, m, n}]/(2^m m!), {m, 1, n}];
%t Table[a[n], {n, 0, 23}] (* _Jean-François Alcover_, Aug 08 2018, from Maxima *)
%t Range[0, 23]! CoefficientList[ Series[ Exp[Sin[x] + 2 Sin[x]^2], {x, 0, 23}], x] (* _Robert G. Wilson v_, Aug 08 2018 *)
%o (Maxima)
%o a(n):=sum(sum((binomial(m, k-m)*((-1)^(n-k)+1)*sum((2*i-k)^n*binomial(k, i)*(-1)^((n+k)/2-i), i, 0, k/2)), k, m, n)/(2^m*m!), m, 1, n);
%o (PARI)
%o x='x+O('x^66); /* that many terms */
%o Vec(serlaplace(exp(sin(x)+2*sin(x)^2))) /* show terms */ /* _Joerg Arndt_, Apr 25 2011 */
%K sign
%O 0,3
%A _Vladimir Kruchinin_, Apr 21 2011
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