A003710 Expansion of e.g.f. cos(tan(x)) (even powers only).

1, -1, -7, -97, -2063, -53409, -752343, 166831871, 43685848289, 9398558916159, 2116926930779225, 524586454143030495, 144620290378876829905, 44287070229737735633567, 14954349885478653319004041

Offset: 0

Author

R. H. Hardin, Simon Plouffe

Keywords

• sign

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A003709, A003711.

Programs

Mathematica

nn = 20; Table[(CoefficientList[Series[Cos[Tan[x]], {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* Vaclav Kotesovec, Feb 16 2015 *)

Maxima

a(n):=if n=0 then 1 else 2*sum((sum(binomial(j-1,2*m-1)*j!*2^(2*n-j-1)*(-1)^(n+j)*stirling2(2*n,j),j,2*m,2*n))/(2*m)!,m,0,n); /* Vladimir Kruchinin, Jun 28 2011 */

PARI

my(x='x+O('x^30)); select(x->x, Vec(serlaplace(cos(tan(x))))) \\ Michel Marcus, Oct 02 2021

Formula

a(n) = 2 * Sum_{m=0..n} ( Sum_{j=2*m..2*n} binomial(j-1,2*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 29 2011