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%I #7 Dec 18 2017 22:51:34
%S 1,3,53,2359,198953,27412011,5625656541,1613676694239,617477049181521,
%T 304167421243513683,187546541676182230149,141512355477854459198343,
%U 128265950128144233675269241,137512081213377707268891639675,172108297920263623816775456321325
%N Expansion of e.g.f. arcsin(arctanh(x)) (odd powers only).
%H Robert Israel, <a href="/A296680/b296680.txt">Table of n, a(n) for n = 0..215</a>
%F E.g.f.: arcsinh(arctan(x)) (odd powers only, absolute values).
%F E.g.f.: -i*log((i/2)*(log(1 + x) - log(1 - x)) + sqrt(1 - (log(1 + x) - log(1 - x))^2/4)), where i is the imaginary unit (odd powers only).
%e arcsin(arctanh(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2359*x^7/7! + 198953*x^9/9! + 27412011*x^11/11! + ...
%p S:= series(arcsin(arctanh(x)),x,52):
%p seq(coeff(S,x,n)*n!,n=1..51,2); # _Robert Israel_, Dec 18 2017
%t nmax = 15; Table[(CoefficientList[Series[ArcSin[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
%t nmax = 15; Table[(CoefficientList[Series[-I Log[(I/2) (Log[1 + x] - Log[1 - x]) + 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, A003717, A010050, A012134, A296464, A296466, A296679.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Dec 18 2017
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