OFFSET
1,2
COMMENTS
Note that arctanh(x) = log((1+x)/(1-x))/2.
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
EXAMPLE
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.
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[Tanh[x]/E^x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 10 2014 *)
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 14 2013
STATUS
approved