A216401 E.g.f.: artanh(x*exp(x)).

1, 2, 5, 28, 209, 1806, 18997, 235544, 3322881, 52688890, 929147141, 18023207412, 381330466321, 8740727309510, 215767934510325, 5706703994412976, 160994795504231297, 4825786400923162482, 153160894479441852037, 5131078462229088189260

Offset: 1

Author

Paul D. Hanna, Sep 06 2012

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 5*x^3/3! + 28*x^4/4! + 209*x^5/5! + 1806*x^6/6! + ...
such that tanh(A(x)) = x*exp(x) and
A(x) = exp(x)*x + exp(3*x)*x^3/3 + exp(5*x)*x^5/5 + exp(7*x)*x^7/7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Wikipedia contributors, Area function (inverse hyperbolic function), Wikipedia, the free encyclopedia, as of April 7, 2025.

Programs

Mathematica

CoefficientList[Series[Log[(1+x*E^x)/(1-x*E^x)]/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 12 2013 *)

PARI

{a(n)=n!*polcoeff(atanh(x*exp(x +x*O(x^n))),n)}

PARI

{a(n)=n!*sum(k=0,(n-1)\2,(2*k+1)^(n-2*k-2)/(n-2*k-1)!)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f.: Sum_{n>=1} exp((2*n-1)*x) * x^(2*n-1) / (2*n-1).

E.g.f.: log( (1+x*exp(x)) / (1-x*exp(x)) ) / 2.

a(n) = n! * Sum_{k=0..floor((n-1)/2)} (2*k+1)^(n-2*k-2) / (n-2*k-1)!.

a(n) ~ (n-1)!/(2*LambertW(1)^n). - Vaclav Kotesovec, Feb 12 2013