A381276 Expansion of e.g.f. exp(x * cos(3*x)).

1, 1, 1, -26, -107, 136, 9181, 53488, -427895, -10486016, -43859879, 1373548672, 23512856797, -30564574208, -6412871847563, -73709639926784, 1060067525174929, 40587133606543360, 179320588932698929, -14474677657838059520, -306563699887974043739, 2301792469199499132928

Offset

0,4

Comments

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

Links

Harvey P. Dale, Table of n, a(n) for n = 0..473

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-9)^k * (2*k+1) * binomial(n-1,2*k) * a(n-2*k-1).

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

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[x Cos[3x]], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jan 05 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, (3*I)^(n-k)*a185951(n, k));

Crossrefs

Cf. A009189, A381275.

Cf. A185951, A352643, A381283.

Sequence in context: A042320 A042322 A042324 * A044277 A044658 A024381

Adjacent sequences: A381273 A381274 A381275 * A381277 A381278 A381279

Keyword

sign

Author

Seiichi Manyama, Feb 18 2025

Status

approved