A352252 Expansion of e.g.f. 1 / (1 - x * cos(x)).

1, 1, 2, 3, 0, -55, -480, -3157, -15232, -16623, 898560, 16316179, 194574336, 1666248025, 5418649600, -170157839685, -5164467978240, -92955464490463, -1188910801354752, -7329026447550685, 157257042777866240, 7516793832172469481, 187200588993188069376

Offset: 0

Author

Ilya Gutkovskiy, Mar 09 2022

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..456

Crossrefs

Cf. A000111, A007289, A009189, A094088, A205571, A302397, A352250, A352251.

Programs

Mathematica

nmax = 22; CoefficientList[Series[1/(1 - x Cos[x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[(-1)^k Binomial[n, 2 k + 1] (2 k + 1) a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 22}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(1 / (1 - x * cos(x)))) \\ Michel Marcus, Mar 10 2022

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, k!*I^(n-k)*a185951(n, k)); \\ Seiichi Manyama, Feb 17 2025

Formula

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

a(n) = Sum_{k=0..n} k! * i^(n-k) * A185951(n,k), where i is the imaginary unit. - Seiichi Manyama, Feb 17 2025