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%I #7 Dec 13 2017 18:36:48
%S 1,4,88,4688,459520,71876352,16428530688,5167215464448,
%T 2140879726411776,1130276555155243008,740796870212763254784,
%U 590192778209307913617408,561748717440430309770264576,629564244208933873601143111680,820602153197407426121272991416320,1230877720962045060728502509025361920
%N Expansion of e.g.f. arctanh(arctanh(x)) (odd powers only).
%F E.g.f.: arctan(arctan(x)) (odd powers only, absolute values).
%F E.g.f.: (log(2 - log(1 - x) + log(1 + x)) - log(2 + log(1 - x) - log(1 + x)))/2 (odd powers only).
%F a(n) ~ (2*n)! * ((exp(2) + 1)/(exp(2) - 1))^(2*n+1). - _Vaclav Kotesovec_, Dec 13 2017
%e arctanh(arctanh(x)) = x/1! + 4*x^3/3! + 88*x^5/5! + 4688*x^7/7! + 459520*x^9/9! + ...
%t nmax = 16; Table[(CoefficientList[Series[ArcTanh[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[2 - Log[1 - x] + Log[1 + x]] - Log[2 + Log[1 - x] - Log[1 + x]])/2, {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%Y Cf. A003718, A010050, A012263, A296467.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 13 2017
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