A227466 E.g.f. equals the series reversion of tanh(x) / exp(x).

1, 2, 11, 96, 1149, 17520, 324855, 7096320, 178495065, 5081045760, 161485639875, 5668185600000, 217773547039125, 9090267234048000, 409648199646660975, 19822083757572096000, 1025047834112461784625, 56415976360602034176000, 3292602426688307236378875

Offset: 1

Author

Paul D. Hanna, Jul 14 2013

Keywords

• nonn

Comments

Note that arctanh(x) = log((1+x)/(1-x))/2.

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 11*x^3/3! + 96*x^4/4! + 1149*x^5/5! + 17520*x^6/6! + ...
where A( tanh(x)/exp(x) ) = x.
		

Crossrefs

Cf. A227465.

Programs

Mathematica

Rest[CoefficientList[InverseSeries[Series[Tanh[x]/E^x,{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 10 2014 *)

PARI

{a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(tanh(X)/exp(X)), n)}
for(n=1,25,print1(a(n),", "))

PARI

{a(n)=local(A=x); for(i=1,n,A=atanh(x*exp(A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: A(x) = arctanh(x*exp(A(x))).

a(n) ~ ((1+sqrt(5))/2)^(5*n/2) * n^(n-1) / (5^(1/4) * exp(n)). - Vaclav Kotesovec, Jan 10 2014