A352638 Expansion of e.g.f. 1/(1 - 3*sin(x)).

1, 3, 18, 159, 1872, 27543, 486288, 10016619, 235798272, 6244714443, 183756215808, 5947907121879, 210026879004672, 8034293365747743, 330982609573398528, 14609181655918083939, 687820834029346947072, 34407546247054875367443, 1822450167175258689896448

Offset: 0

Author

Seiichi Manyama, Mar 25 2022

Keywords

• nonn

Crossrefs

Cf. A000111, A007289.

Cf. A107403, A352640.

Cf. A136630.

Programs

Mathematica

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

PARI

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

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)));

Formula

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).

a(n) ~ n! / (2^(3/2) * arcsin(1/3)^(n+1)). - Vaclav Kotesovec, Mar 26 2022

a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025