A296730 Expansion of e.g.f. arctanh(x*cos(x)) (odd powers only).

1, -1, -31, -337, 24705, 2451679, -17936543, -42895630065, -5396647099903, 1239561882325439, 708575518706816481, 37448619025871342959, -113842057082636742446975, -52054011876398495316250977, 16226448322449614832534708065, 31975745831751940004484917311439

Offset: 0

Author

Ilya Gutkovskiy, Dec 19 2017

Keywords

• sign

Examples

			arctanh(x*cos(x)) = x/1! - x^3/3! - 31*x^5/5! - 337*x^7/7! + 24705*x^9/9! + ...
		

Crossrefs

Cf. A000364, A009015, A009016, A009446, A009447, A009633, A009634, A010050, A012494, A191512, A296465, A296467, A296728, A296729, A296731, A296740.

Programs

Mathematica

nmax = 16; Table[(CoefficientList[Series[ArcTanh[x Cos[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*cos(x)).