A296678 Expansion of e.g.f. arctanh(arcsin(x)) (odd powers only).

1, 3, 53, 2303, 185033, 23756667, 4457821821, 1150764459063, 391167511473681, 169370797497060339, 91013260219635394629, 59435772666287730632559, 46362471059282707504957401, 42577231265939498962852834155, 45471686987452309473064526678925

Offset: 0

Author

Ilya Gutkovskiy, Dec 18 2017

Keywords

• nonn

Examples

			arctanh(arcsin(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2303*x^7/7! + 185033*x^9/9! + 23756667*x^11/11! + ...
		

Crossrefs

Cf. A001818, A003716, A010050, A012258, A296465, A296467, A296677.

Programs

Mathematica

nmax = 15; Table[(CoefficientList[Series[ArcTanh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]]/2 - Log[1 + I 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.: arctan(arcsinh(x)) (odd powers only, absolute values).

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