A352647 Expansion of e.g.f. 1/(1 - 3 * x * cos(x)).

1, 3, 18, 153, 1728, 24315, 410400, 8079729, 181786752, 4601232243, 129402385920, 4003157532297, 135098815002624, 4939266681129963, 194472450526169088, 8203835046344538465, 369151362125290045440, 17649035213360472293091, 893431062200523039178752

Offset: 0

Author

Seiichi Manyama, Mar 25 2022

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..384

Crossrefs

Cf. A352252, A352646.

Cf. A352643, A352649.

Cf. A185951.

Programs

Mathematica

With[{m = 18}, Range[0, m]! * CoefficientList[Series[1/(1 - 3*x*Cos[x]), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

PARI

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

PARI

a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (-1)^k*(2*k+1)*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 * (2*k+1) * binomial(n,2*k+1) * a(n-2*k-1).

a(n) = Sum_{k=0..n} 3^k * k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 25 2025