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

0,3

Comments

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

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A014307.

Sequence in context: A204190 A051295 A334154 * A104429 A109319 A059219

Adjacent sequences: A009380 A009381 A009382 * A009384 A009385 A009386

Keyword

sign,easy

Author

R. H. Hardin

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Status

approved