A352642 Expansion of e.g.f. exp(2 * x * cos(x)).

1, 2, 4, 2, -32, -198, -416, 2634, 30720, 107378, -605696, -10282094, -46020608, 304968874, 6121832448, 29994597338, -279697555456, -5729595393310, -26849178681344, 401845799334690, 7714801999937536, 29062583111892506, -812705956979802112

Offset: 0

Author

Seiichi Manyama, Mar 25 2022

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..545

Crossrefs

Cf. A009189, A185951, A352643.

Programs

Mathematica

With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[2*x*Cos[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*x*cos(x))))

PARI

a(n) = if(n==0, 1, 2*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) = 2 * 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} 2^k * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 18 2025