A381275 Expansion of e.g.f. exp(x * cos(2*x)).

1, 1, 1, -11, -47, -39, 1681, 10893, -13215, -851471, -5515679, 34375397, 887687857, 3982645577, -85350572943, -1466457337859, -659043831871, 270733024430305, 3181606182917569, -24432689736388395, -1076204061663657839, -6834631528147762247, 221729710998069153617

Offset

0,4

Comments

As stated in the comment of A185951, A185951(n,0) = 0^n.

Links

Table of n, a(n) for n=0..22.

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).

a(n) = Sum_{k=0..n} (2*i)^(n-k) * A185951(n,k), where i is the imaginary unit.

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[x Cos[2x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 07 2026 *)

Prog

(PARI) a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (2*I)^(n-k)*a185951(n, k));

Crossrefs

Cf. A009189, A381276.

Cf. A185951.

Sequence in context: A361888 A179786 A238584 * A033209 A107216 A158809

Adjacent sequences: A381272 A381273 A381274 * A381276 A381277 A381278

Keyword

sign

Author

Seiichi Manyama, Feb 18 2025

Status

approved