A296465 Expansion of e.g.f. arctanh(arctanh(x)) (odd powers only).

1, 4, 88, 4688, 459520, 71876352, 16428530688, 5167215464448, 2140879726411776, 1130276555155243008, 740796870212763254784, 590192778209307913617408, 561748717440430309770264576, 629564244208933873601143111680, 820602153197407426121272991416320, 1230877720962045060728502509025361920

Offset: 0

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

• nonn

Examples

			arctanh(arctanh(x)) = x/1! + 4*x^3/3! + 88*x^5/5! + 4688*x^7/7! + 459520*x^9/9! + ...
		

Crossrefs

Cf. A003718, A010050, A012263, A296467.

Programs

Mathematica

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

Formula

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

E.g.f.: (log(2 - log(1 - x) + log(1 + x)) - log(2 + log(1 - x) - log(1 + x)))/2 (odd powers only).

a(n) ~ (2*n)! * ((exp(2) + 1)/(exp(2) - 1))^(2*n+1). - Vaclav Kotesovec, Dec 13 2017