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Expansion of e.g.f. arctan(arcsin(x)) (odd powers only).
2

%I #5 Dec 18 2017 19:43:17

%S 1,-1,13,-173,12409,-370137,88556037,-2668274373,2491377242481,

%T 34526890553679,202383113207336829,25792743610973373219,

%U 39172126704113226631401,12501799823936578879327095,15717805122762984314778029685,9078237580992214462785729689355

%N Expansion of e.g.f. arctan(arcsin(x)) (odd powers only).

%F E.g.f.: (i/2)*log(1 - log(i*x + sqrt(1 - x^2))) - (i/2)*log(1 + log(i*x + sqrt(1 - x^2))), where i is the imaginary unit (odd powers only).

%e arctan(arcsin(x)) = x/1! - x^3/3! + 13*x^5/5! - 173*x^7/7! + 12409*x^9/9! - 370137*x^11/11! + ...

%t nmax = 16; Table[(CoefficientList[Series[ArcTan[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

%t nmax = 16; Table[(CoefficientList[Series[(I/2) Log[1 - Log[I x + Sqrt[1 - x^2]]] - (I/2) Log[1 + 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, A003705, A010050, A012194, A096719, A096720, A296465, A296467.

%K sign

%O 0,3

%A _Ilya Gutkovskiy_, Dec 18 2017