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%I #5 Dec 18 2017 11:54:34
%S 0,2,-12,328,-15008,1356192,-166628352,31500831360,-7474571071488,
%T 2418220114014720,-940432709166170112,464609611973533501440,
%U -268355615175956213268480,188067307050238642631516160,-151072053399934628129585233920,142618740583722182161589570273280
%N Expansion of e.g.f. log(1 + arcsin(x)*arcsinh(x)) (even powers only).
%F E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2))*log(x + sqrt(1 + x^2))), where i is the imaginary unit (even powers only).
%e log(1 + arcsin(x)*arcsinh(x)) = 2*x^2/2! - 12*x^4/4! + 328*x^6/6! - 15008*x^8/8! + 1356192*x^10/10! - 166628352*x^12/12! + ...
%t nmax = 15; Table[(CoefficientList[Series[Log[1 + ArcSin[x] ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%t nmax = 15; Table[(CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]] Log[x + Sqrt[1 + x^2]]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
%Y Cf. A001818, A009232, A009359, A012598, A189815, A296435.
%K sign
%O 0,2
%A _Ilya Gutkovskiy_, Dec 17 2017
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