%I M4453 #36 Oct 02 2021 07:56:28
%S 1,-1,-7,-97,-2063,-53409,-752343,166831871,43685848289,9398558916159,
%T 2116926930779225,524586454143030495,144620290378876829905,
%U 44287070229737735633567,14954349885478653319004041
%N Expansion of e.g.f. cos(tan(x)) (even powers only).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A003710/b003710.txt">Table of n, a(n) for n = 0..100</a>
%F 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
%t 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 *)
%o (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 */
%o (PARI) my(x='x+O('x^30)); select(x->x, Vec(serlaplace(cos(tan(x))))) \\ _Michel Marcus_, Oct 02 2021
%Y Cf. A003709, A003711.
%K sign
%O 0,3
%A _R. H. Hardin_, _Simon Plouffe_