A296464 Expansion of e.g.f. arcsin(arcsin(x)) (odd powers only).

1, 2, 28, 1024, 71632, 8192736, 1392793920, 330041217024, 104069101383936, 42159457593506304, 21346870862961183744, 13213529766600134344704, 9818417126704155249954816, 8625630408510010165396070400, 8844234850947343105068735283200, 10467364426053362392901751845683200

Offset: 0

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

• nonn

Examples

			arcsin(arcsin(x)) = x/1! + 2*x^3/3! + 28*x^5/5! + 1024*x^7/7! + 71632*x^9/9! + ...
		

Crossrefs

Cf. A001818, A003712, A012063, A296466.

Programs

Mathematica

nmax = 16; Table[(CoefficientList[Series[ArcSin[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 16; Table[(CoefficientList[Series[-I Log[Log[I x + Sqrt[1 - x^2]] + Sqrt[1 + Log[I x + Sqrt[1 - x^2]]^2]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

Formula

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

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

a(n) ~ sqrt(2) * (2*n)! / (sqrt(Pi*sin(2)*n) * sin(1)^(2*n)). - Vaclav Kotesovec, Dec 13 2017