A189420 - OEIS

%I #15 Oct 30 2022 11:03:49

%S 1,1,3,6,13,12,-121,-896,-4391,-10160,64491,900768,6118693,16033344,

%T -198382609,-3101259776,-22263439439,-23508393728,1747001723475,

%U 24367272291840,145393520219965

%N Expansion of e.g.f. exp(sin(x)+sin(x)^2).

%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, (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^k)/m!).

%t With[{nn=20},CoefficientList[Series[Exp[Sin[x]+Sin[x]^2],{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 01 2017 *)

%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))/2^k,k,m,n)/m!,m,1,n);

%K sign

%O 0,3

%A _Vladimir Kruchinin_, Apr 21 2011