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%I #6 Dec 13 2017 18:36:55
%S 1,0,8,56,8000,342144,68623488,8295676416,2411783847936,
%T 584142614728704,240810283258527744,96772676958798741504,
%U 54867909992513301282816,32661008325245409302937600,24691868812821871169667072000,20243305132513358736699378892800,19829947398943934886214249532620800
%N Expansion of e.g.f. arcsinh(arcsin(x)) (odd powers only).
%F E.g.f.: arcsin(arcsinh(x)) (odd powers only, absolute values).
%F E.g.f.: log(sqrt(1 - log(i*x + sqrt(1 - x^2))^2) - i*log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).
%F a(n) ~ 2 * (2*n)! / sqrt(Pi*(4 + Pi^2)*n). - _Vaclav Kotesovec_, Dec 13 2017
%e arcsinh(arcsin(x)) = x/1! + 8*x^5/5! + 56*x^7/7! + 8000*x^9/9! + 342144*x^11/11! + 68623488*x^13/13! + ...
%t nmax = 17; Table[(CoefficientList[Series[ArcSinh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%t nmax = 17; Table[(CoefficientList[Series[Log[Sqrt[1 - Log[I x + Sqrt[1 - x^2]]^2] - I Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%Y Cf. A001818, A003722, A012248, A296464.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Dec 13 2017
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