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%I #21 Jul 24 2018 21:23:36
%S 1,0,-12,-120,-2352,-75840,-1649472,118634880,41344643328,
%T 9528901232640,2213829515240448,559192086549719040,
%U 156367986602421669888,48476425507418343751680,16569615994864645076533248
%N Expansion of e.g.f. cos(tan(x)*sin(x)) (even powers only).
%H G. C. Greubel, <a href="/A009078/b009078.txt">Table of n, a(n) for n = 0..240</a>
%F a(n) = Sum_{k=1..n} (4^(n-k)*Sum_{t=k..n-k} binomial(2*n,2*t)*((Sum_{j=2*k..2*n-2*t} binomial(j-1,2*k-1)*j!*stirling2(2*n-2*t,j)*(-1)^(n+j)*2^(1-j)))*sum(i=0..k, (i-k)^(2*t)*binomial(2*k,i)*(-1)^(k-i)))/(2*k)!, n>0, a(0)=1. - _Vladimir Kruchinin_, Jun 30 2011
%t With[{nn=30},Take[CoefficientList[Series[Cos[Tan[x]Sin[x]],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Jun 04 2018 *)
%o (Maxima)
%o a(n):=if n=0 then 1 else sum((4^(n-k)*sum(binomial(2*n,2*t)*((sum(binomial(j-1,2*k-1)*j!*stirling2(2*n-2*t,j)*(-1)^(n+j)*2^(1-j),j,2*k,2*n-2*t))*sum((i-k)^(2*t)*binomial(2*k,i)*(-1)^(k-i),i,0,k)),t,k,n-k))/(2*k)!,k,1,n); /* _Vladimir Kruchinin_, Jun 30 2011 */
%o (PARI) x='x+O('x^50); v=Vec(serlaplace(cos(tan(x)*sin(x)))); vector(#v\2,n,v[2*n-1]) \\ _G. C. Greubel_, Jul 24 2018
%K sign
%O 0,3
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
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