|
%I #10 Mar 10 2022 08:27:20
%S 1,2,20,486,21944,1591210,169207092,24808395262,4796420822384,
%T 1182349445882706,361939981107422060,134705596642758848806,
%U 59900689507397744253096,31365504832631796986962426,19102102945852191813235300004,13387748268024668296590660222030
%N Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).
%F a(0) = 1; a(n) = 2 * Sum_{k=1..n} (-1)^(k+1) * binomial(2*n,2*k) * k * a(n-k).
%t nmax = 30; Take[CoefficientList[Series[1/(1 - x Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
%t a[0] = 1; a[n_] := a[n] = 2 Sum[(-1)^(k + 1) Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
%o (PARI) my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ _Michel Marcus_, Mar 10 2022
%Y Cf. A000111, A000364, A009214, A205571, A210657, A302397, A352251, A352252.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Mar 09 2022
|
