%I #5 Dec 18 2017 19:43:32
%S 1,1,13,341,18649,1599849,205524837,36391450941,8546308276401,
%T 2564025898856913,957697868873929149,435619128300038521893,
%U 237104370189582892175241,152148421079949399306125625,113672892845152570858515803925,97820056722556900357454981990925
%N Expansion of e.g.f. arcsinh(arctanh(x)) (odd powers only).
%F E.g.f.: arcsin(arctan(x)) (odd powers only, absolute values).
%F E.g.f.: log((log(1 + x) - log(1 - x))/2 + sqrt(1 + (log(1 + x) - log(1 - x))^2/4)) (odd powers only).
%e arcsinh(arctanh(x)) = x/1! + x^3/3! + 13*x^5/5! + 341*x^7/7! + 18649*x^9/9! + 1599849*x^11/11! + ...
%t nmax = 16; Table[(CoefficientList[Series[ArcSinh[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%t nmax = 16; Table[(CoefficientList[Series[Log[(Log[1 + x] - Log[1 - x])/2 + Sqrt[1 + (Log[1 + x] - Log[1 - x])^2/4]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%Y Cf. A001818, A003706, A010050, A012254, A096717, A096718, A296464, A296466, A296677.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 18 2017