A012278 Expansion of e.g.f. arctanh(exp(x)*log(x+1)).

0, 1, 1, 4, 12, 103, 595, 6508, 58800, 786973, 9744373, 155024956, 2434745852, 45189575715, 856361535783, 18256766891140, 403804360914560, 9755015402674937, 246067759361332137, 6656604425348335060, 188304809071878207052, 5645851709034522319007

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..422 (terms 0..200 from Alois P. Heinz)

Formula

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

Example

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

Maple

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

Mathematica

CoefficientList[Series[ArcTanh[Exp[x]*Log[x + 1]], {x, 0, 20}], x]*
Range[0, 20]! (* Bruno Berselli, Feb 17 2013 *)

Prog

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

Crossrefs

Sequence in context: A038053 A217155 A120267 * A070040 A079822 A216043

Adjacent sequences: A012275 A012276 A012277 * A012279 A012280 A012281

Keyword

nonn

Author

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

Extensions

Prepended a(0)=0 by Bruno Berselli, Feb 17 2013

Status

approved