A296467 Expansion of e.g.f. arctan(arctanh(x)) (odd powers only).

1, 0, 8, 112, 8192, 599808, 80010240, 13537247232, 3160676007936, 929451393220608, 343173318976733184, 154043745649772986368, 82935056810462020632576, 52660879605487383997317120, 38970318170642827020431523840, 33236188662933234332228627988480, 32365907321554306913981616441262080

Offset: 0

Author

Ilya Gutkovskiy, Dec 13 2017

Keywords

• nonn

Examples

			arctan(arctanh(x)) =  x/1! + 8*x^5/5! + 112*x^7/7! + 8192*x^9/9! + 599808*x^11/11! + 80010240*x^13/13! + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..224

Crossrefs

Cf. A003721, A010050, A012231, A296465.

Programs

Maple

S:= series(arctan(arctanh(x)),x,52):
seq(coeff(S,x,2*i+1)*(2*i+1)!,i=0..25); # Robert Israel, Dec 13 2017

Mathematica

nmax = 17; Table[(CoefficientList[Series[ArcTan[ArcTanh[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 17; Table[(CoefficientList[Series[I (Log[2 + I Log[1 - x] - I Log[1 + x]] - Log[2 - I Log[1 - x] + I 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.: arctanh(arctan(x)) (odd powers only, absolute values).

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