%I #4 Dec 26 2017 18:06:18

%S 0,1,1,-12,-193,5195,397248,-9589391,-3147743231,-10931156748,

%T 65632780196255,4713930109297211,-2846093176389647904,

%U -606335605925899344287,213167747093485780707937,109460864600185764327567060,-21782399212761670190907400897

%N Expansion of e.g.f. arcsinh(x*tan(x/2)) (even powers only).

%F a(n) = (2*n)! * [x^(2*n)] arcsinh(x*tan(x/2)).

%e arcsinh(x*tan(x/2)) = x^2/2! + x^4/4! - 12*x^6/6! - 193*x^8/8! + ...

%t nmax = 16; Table[(CoefficientList[Series[ArcSinh[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

%Y Cf. A000364, A001469, A001818, A110501, A296839, A296841, A296842, A296853, A296854, A296856, A296939, A296940, A296941.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Dec 22 2017