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%I #6 Dec 21 2017 06:07:25
%S 1,2,20,504,24464,1959840,234852672,39370660224,8799246209280,
%T 2528787321598464,908585701684024320,399070678264750356480,
%U 210373049449102957645824,131083661069772517440921600,95304505860052894815543705600,79961055068441273887848131297280
%N Expansion of e.g.f. exp(x*arctanh(x)) (even powers only).
%F a(n) = (2*n)! * [x^(2*n)] exp(x*arctanh(x)).
%F a(n) ~ 2^(2*n + 2) * n^(2*n) / exp(2*n). - _Vaclav Kotesovec_, Dec 21 2017
%e exp(x*arctanh(x)) = 1 + 2*x^2/2! + 20*x^4/4! + 504*x^6/6! + 24464*x^8/8! + ...
%t nmax = 15; Table[(CoefficientList[Series[Exp[x ArcTanh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%t nmax = 15; Table[(CoefficientList[Series[Exp[x (Log[1 + x] - Log[1 - x])/2], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A000246, A009252, A009273, A010050, A166356, A259647, A293193, A296787, A296788.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 20 2017