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Expansion of e.g.f. sin(x)*sin(sin(x)) (even powers only).
2

%I #26 May 02 2024 14:45:39

%S 0,2,-12,118,-1816,37354,-974372,31769182,-1259350576,59073098706,

%T -3226127944764,202778723085382,-14503292667068744,

%U 1168138817072817594,-105070093531389641300,10481640556778901446190,-1152522131016352310551648

%N Expansion of e.g.f. sin(x)*sin(sin(x)) (even powers only).

%H G. C. Greubel, <a href="/A009546/b009546.txt">Table of n, a(n) for n = 0..250</a>

%F a(n) = 2*Sum_{k=1..n} (4^(n-k)*Sum_{i=0..k} (i-k)^(2*n)*binomial(2*k,i)*(-1)^(n-i-1))/(2*k-1)!). - _Vladimir Kruchinin_, Jun 28 2011

%t With[{nmax = 50}, CoefficientList[Series[Sin[x]*Sin[Sin[x]], {x,0,nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]] (* _G. C. Greubel_, Jan 20 2018 *)

%o (Maxima)

%o a(n):=2*sum((4^(n-k)*sum((i-k)^(2*n)*binomial(2*k,i)*(-1)^(n-i-1),i,0,k))/(2*k-1)!,k,1,n); /* _Vladimir Kruchinin_, Jun 28 2011 */

%o (PARI) x='x+O('x^50); v=Vec(serlaplace(sin(x)*sin(sin(x)))); concat([0], vector(#v\2,n,v[2*n-1])) \\ _G. C. Greubel_, Jan 20 2018

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier GĂ©rard_, Mar 15 1997