A296466 Expansion of e.g.f. arcsinh(arcsin(x)) (odd powers only).

1, 0, 8, 56, 8000, 342144, 68623488, 8295676416, 2411783847936, 584142614728704, 240810283258527744, 96772676958798741504, 54867909992513301282816, 32661008325245409302937600, 24691868812821871169667072000, 20243305132513358736699378892800, 19829947398943934886214249532620800

Offset: 0

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

• nonn

Examples

			arcsinh(arcsin(x)) = x/1! + 8*x^5/5! + 56*x^7/7! + 8000*x^9/9! + 342144*x^11/11! + 68623488*x^13/13! + ...
		

Crossrefs

Cf. A001818, A003722, A012248, A296464.

Programs

Mathematica

nmax = 17; Table[(CoefficientList[Series[ArcSinh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[Log[Sqrt[1 - Log[I x + Sqrt[1 - x^2]]^2] - I Log[I x + Sqrt[1 - x^2]]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

E.g.f.: arcsin(arcsinh(x)) (odd powers only, absolute values).

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

a(n) ~ 2 * (2*n)! / sqrt(Pi*(4 + Pi^2)*n). - Vaclav Kotesovec, Dec 13 2017