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%I #6 Dec 13 2017 18:36:33
%S 0,2,4,158,3624,427482,29665260,6948032310,991515848400,
%T 383952670412850,93532380775766100,53913667654307868750,
%U 20087427376748637675000,16096655588343149442026250,8531309209053208518037597500,9057367559484733295974741323750,6486329752640392315697926589700000
%N Expansion of e.g.f. arcsinh(x)*arctanh(x) (even powers only).
%F E.g.f.: arcsin(x)*arctan(x) (even powers only, absolute values).
%F E.g.f.: (log(1 + x) - log(1 - x))*log(x + sqrt(1 + x^2))/2 (even powers only).
%F a(n) ~ (2*n-1)! * log(1+sqrt(2)) * (1 - (-1)^n * sqrt(Pi) / (4 * log(1+sqrt(2)) * sqrt(n))). - _Vaclav Kotesovec_, Dec 13 2017
%e arcsinh(x)*arctanh(x) = 2*x^2/2! + 4*x^4/4! + 158*x^6/6! + 3624*x^8/8! + 427482*x^10/10! + ..
%t nmax = 16; Table[(CoefficientList[Series[ArcSinh[x] ArcTanh[x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%t nmax = 16; Table[(CoefficientList[Series[(Log[1 + x] - Log[1 - x]) Log[x + Sqrt[1 + x^2]]/2, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A001818, A009744, A009747, A010050, A012752, A296462.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 13 2017
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