%I #28 Sep 08 2022 08:44:38

%S 0,1,1,4,12,103,595,6508,58800,786973,9744373,155024956,2434745852,

%T 45189575715,856361535783,18256766891140,403804360914560,

%U 9755015402674937,246067759361332137,6656604425348335060,188304809071878207052,5645851709034522319007

%N Expansion of e.g.f. arctanh(exp(x)*log(x+1)).

%H G. C. Greubel, <a href="/A012278/b012278.txt">Table of n, a(n) for n = 0..422</a> (terms 0..200 from Alois P. Heinz)

%F a(n) ~ (n-1)! / (2*(exp(r)-1)^n), where r = 0.5122224330332299... is the root of the equation r*exp(exp(r)-1)=1. - _Vaclav Kotesovec_, Oct 24 2013

%e arctanh(exp(x)*log(x+1)) = x+1/2!*x^2+4/3!*x^3+12/4!*x^4+103/5!*x^5...

%p seq(coeff(series(factorial(n)*arctanh(exp(x)*log(x+1)),x,n+1), x, n), n = 0 .. 22); # _Muniru A Asiru_, Oct 28 2018

%t CoefficientList[Series[ArcTanh[Exp[x]*Log[x + 1]], {x, 0, 20}], x]*

%t Range[0, 20]! (* _Bruno Berselli_, Feb 17 2013 *)

%o (PARI) x='x+O('x^30); concat([0], Vec(serlaplace(atanh(exp(x)* log(x+1))))) \\ _G. C. Greubel_, Oct 28 2018

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Argtanh(Exp(x)*Log(x+1)) )); [0] cat [Factorial(n)*b[n]: n in [1..m-1]]; // _G. C. Greubel_, Oct 28 2018

%K nonn

%O 0,4

%A Patrick Demichel (patrick.demichel(AT)hp.com)

%E Prepended a(0)=0 by _Bruno Berselli_, Feb 17 2013