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%I #23 Jun 23 2017 20:08:45
%S 1,2,13,80,721,7232,87613,1193600,18431041,315130112,5936395213,
%T 121904337920,2712867153361,65005861947392,1669061195146813,
%U 45709411009495040,1330071346341607681,40979304201240707072,1332713597815933766413,45623097880495676456960
%N E.g.f.: sinh(x)/(1-sinh(x)-sinh(x)^2).
%H Robert Israel, <a href="/A189087/b189087.txt">Table of n, a(n) for n = 1..413</a>
%F a(n) = sum(k=1..n, sum(i=0..k, (-1)^i*(k-2*i)^n * binomial(k,i)) / (2^k) * A000045(k)).
%F a(n) ~ n! * sqrt(5-sqrt(5))/(5*sqrt(2))/(arcsinh(sqrt(5)/2-1/2))^(n+1). - _Vaclav Kotesovec_, Jun 27 2013
%p h:= sinh(x)/(1-sinh(x)-sinh(x)^2):
%p S:= series(h, x, 61):
%p seq(coeff(S,x,j)*j!, j=1..60); # _Robert Israel_, Jun 23 2017
%t CoefficientList[Series[Sinh[x]/(1-Sinh[x]-Sinh[x]^2), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 27 2013 *)
%o (Maxima)
%o a(n):=sum(sum((-1)^i*(k-2*i)^n*binomial(k,i),i,0,k)/(2^k)*fib(k),k,1,n);
%Y Cf. A000045.
%K nonn
%O 1,2
%A _Vladimir Kruchinin_, Apr 20 2011
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