A009083 Expansion of e.g.f. cos(tan(x)^2) (even powers only).

1, 0, -12, -480, -22512, -1224960, -61017792, 1438993920, 1844639547648, 677206700482560, 225542012911531008, 76252348319434383360, 26581103125260630233088, 9309180001030233433374720

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..236

Programs

Mathematica

With[{nmax = 60}, CoefficientList[Series[Cos[Tan[x]^2], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; -1 ;; 2]] (* G. C. Greubel, Jul 24 2018 *)

Maxima

a(n):=2*sum(((-1)^(m)*sum(binomial(j-1,4*m-1)*j!*2^(2*n-j-1)*(-1)^(n+j)*stirling2(2*n,j),j,4*m,2*n))/(2*m)!,m,0,n); /* Vladimir Kruchinin, Jun 11 2011 */

PARI

x='x+O('x^60); v=Vec(serlaplace(cos(tan(x)^2))); vector(#v\2,n,v[2*n-1]) \\ G. C. Greubel, Jul 24 2018

Formula

a(n) = 2*Sum_{m=0..n} ((-1)^(m)*Sum_{j=4*m..2*n} binomial(j-1,4*m-1)*j!*2^(2*n-j-1)*(-1)^(n+j)*stirling2(2*n,j))/(2*m)!, n>0, a(0)=1. - Vladimir Kruchinin, Jun 11 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997