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%I #4 Dec 21 2017 14:56:35
%S 1,1,2,3,-3,20,105,-5271,133826,-2714517,25525845,2131781300,
%T -235250824479,17527695547713,-1124258412169438,58383380825728035,
%U -975024061456732035,-398903577787777972396,97649546210035758250281,-17069419358223320552890167
%N Expansion of e.g.f. exp(x*tanh(x/2)) (even powers only).
%F a(n) = (2*n)! * [x^(2*n)] exp(x*tanh(x/2)).
%e exp(x*tanh(x/2)) = 1 + x^2/2! + 2*x^4/4! + 3*x^6/6! - 3*x^8/8! + ...
%t nmax = 19; Table[(CoefficientList[Series[Exp[x Tanh[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A001469, A003723, A006229, A009252, A009273, A110501, A296835.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Dec 21 2017