%I #39 Jun 01 2019 09:34:44

%S 0,1,-2,5,-15,54,-240,1350,-9450,78120,-725760,7371000,-81081000,

%T 965487600,-12454041600,173675502000,-2605132530000,41763850128000,

%U -711374856192000,12817252047600000,-243527788904400000

%N Expansion of log(1+tanh(log(1+x))).

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

%H Seiichi Manyama, <a href="/A009383/b009383.txt">Table of n, a(n) for n = 0..450</a>

%F 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

%F a(n) ~ 2 * (n-1)! * (-1)^(n+1). - _Vaclav Kotesovec_, Jan 23 2015

%F 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

%e 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

%t With[{nn=30},CoefficientList[Series[Log[1+Tanh[Log[1+x]]],{x,0,nn}], x]Range[0,nn]!] (* _Harvey P. Dale_, Jan 27 2012 *)

%t CoefficientList[Series[Log[2 - 2/(2 + x*(2 + x))], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 23 2015 *)

%o (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

%Y Cf. A014307.

%K sign,easy

%O 0,3

%A _R. H. Hardin_

%E Extended with signs by _Olivier Gérard_, Mar 15 1997