A296743 Expansion of e.g.f. arctanh(x*sec(x)) (odd powers only).

1, 5, 109, 5977, 612729, 100954061, 24395453861, 8128143367905, 3571195811862385, 2000535014776893973, 1391684597704875555165, 1177047158822263838854889, 1189444022487013498606939625, 1415364934488337503351305867997, 1958850511524588636608881908473749

Offset: 0

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

• nonn

Examples

			arctanh(x*sec(x)) = x/1! + 5*x^3/3! + 109*x^5/5! + 5977*x^7/7! + 612729*x^9/9! + ...
		

Crossrefs

Cf. A003700, A009118, A009119, A009562, A009563, A009765, A009843, A010050, A102075, A191003, A296465, A296467, A296741, A296742.

Programs

Mathematica

nmax = 15; Table[(CoefficientList[Series[ArcTanh[x Sec[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

PARI

first(n) = x='x+O('x^(2*n)); vecextract(Vec(serlaplace(atanh(x/cos(x)))), (4^n - 1)/3) \\ Iain Fox, Dec 19 2017

Formula

a(n) = (2*n+1)! * [x^(2*n+1)] arctanh(x*sec(x)).