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Expansion of e.g.f. 1/(1 - 3*sin(x)).
1

%I #17 Mar 26 2022 13:41:09

%S 1,3,18,159,1872,27543,486288,10016619,235798272,6244714443,

%T 183756215808,5947907121879,210026879004672,8034293365747743,

%U 330982609573398528,14609181655918083939,687820834029346947072,34407546247054875367443

%N Expansion of e.g.f. 1/(1 - 3*sin(x)).

%F a(0) = 1; a(n) = 3 * Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n,2*k+1) * a(n-2*k-1).

%F a(n) ~ n! / (2^(3/2) * arcsin(1/3)^(n+1)). - _Vaclav Kotesovec_, Mar 26 2022

%t With[{m = 17}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*Sin[x]), {x, 0, m}], x]] (* _Amiram Eldar_, Mar 26 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*sin(x))))

%o (PARI) a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*binomial(n, 2*k+1)*a(n-2*k-1)));

%Y Cf. A000111, A007289.

%Y Cf. A107403, A352640.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 25 2022