A009118 Expansion of e.g.f. cos(x/cos(x)) (even powers only).

1, -1, -11, -181, -3863, -66121, 4478365, 1211701763, 226423491793, 43068302925551, 8876725117679941, 1997577117130009403, 483811389670392875449, 121594250947356501211559, 28960468994349845642813677

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..240

Programs

Mathematica

With[{nmax = 60}, CoefficientList[Series[Cos[x/Cos[x]], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* G. C. Greubel, Jul 26 2018 *)

Maxima

a(n):=2*sum(binomial(2*n, 2*m)*sum(binomial(m+j-1, j)*4^(n-m-j)*sum((i-j)^(2*n-2*m)*binomial(2*j, i)*(-1)^(n+j-i), i, 0, j), j, 0, n-m), m, 1, n-1)+(-1)^n; /* Vladimir Kruchinin, Jun 28 2011 */

PARI

x='x+O('x^50); v=Vec(serlaplace(cos(x/cos(x)))); vector(#v\2,n,v[2*n-1]) \\ G. C. Greubel, Jul 26 2018

Formula

a(n) = 2*Sum_{m=1..n-1} binomial(2*n,2*m)*Sum_{j=0..n-m} binomial(m+j-1,j)*4^(n-m-j)*Sum_{i=0..j} (i-j)^(2*n-2*m)*binomial(2*j,i)*(-1)^(n+j-i) +(-1)^n. - Vladimir Kruchinin, Jun 28 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997