A352643 Expansion of e.g.f. exp(3 * x * cos(x)).

1, 3, 9, 18, -27, -552, -3051, -3504, 102825, 1043712, 3192129, -41548416, -653192883, -3033406464, 31367500173, 670266381312, 3916411302609, -40460110970880, -1038593550985479, -6810646726410240, 82445831323038261, 2280185182260854784, 15300402721484153733

Offset: 0

Author

Seiichi Manyama, Mar 25 2022

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..536

Crossrefs

Cf. A009189, A185951, A352642.

Programs

Mathematica

With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[3*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(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-1, 2*k)*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-1,2*k) * a(n-2*k-1).

a(n) = Sum_{k=0..n} 3^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025