A009383 Expansion of log(1+tanh(log(1+x))).

0, 1, -2, 5, -15, 54, -240, 1350, -9450, 78120, -725760, 7371000, -81081000, 965487600, -12454041600, 173675502000, -2605132530000, 41763850128000, -711374856192000, 12817252047600000, -243527788904400000

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

Also expansion of e.g.f. log(1/(1 + Sum_{k>=1} (k+1)/2 * (-x)^k)). - Seiichi Manyama, Jun 01 2019

Examples

			log(1/(1 + Sum_{k>=1} (k+1)/2 * (-x)^k)) = x - 2*x^2/2! + 5*x^3/3! - 15*x^4/4! + 54*x^5/5! - 240*x^6/6! + 1350*x^7/7! - 9450*x^8/8! + ... . - _Seiichi Manyama_, Jun 01 2019
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A014307.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Log[1+Tanh[Log[1+x]]],{x,0,nn}], x]Range[0,nn]!] (* Harvey P. Dale, Jan 27 2012 *)
CoefficientList[Series[Log[2 - 2/(2 + x*(2 + x))], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 23 2015 *)

PARI

{a(n) = if (n<1, 0, -((-1)^n*(n+1)!+sum(k=1, n-1, binomial(n-1, k)*(-1)^k*(k+1)!*a(n-k)))/2)} \\ Seiichi Manyama, Jun 01 2019

Formula

The e.g.f. equals log(2(x+1)^2/(x^2+2x+2)), which has compositional inverse sqrt(exp(x)/(2-exp(x))) - 1. See A014307. - Peter Bala, Mar 23 2013

a(n) ~ 2 * (n-1)! * (-1)^(n+1). - Vaclav Kotesovec, Jan 23 2015

a(n) = (-1/2) * ((-1)^n * (n+1)! + Sum_{k=1..n-1} binomial(n-1,k) * (-1)^k * (k+1)! * a(n-k)). - Seiichi Manyama, Jun 01 2019

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997