OFFSET
1,3
COMMENTS
Compare to: Series_Reversion(log(1+x)*(1+x)) = Sum_{n>=1} -(n-1)^(n-1)*(-x)^n/n!.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
E.g.f. A(x) satisfies: 1 + A(x) = exp( x / sqrt(1 - A(x)^2) ).
a(n) ~ n^(n-1) * (s/((1-s)*sqrt(1-s^2)))^n * ((1-s^2)/sqrt(1+s+s^2)) / exp(n), where s = 0.66288612806066020129... is the root of the equation s*log(1+s) = 1-s. - Vaclav Kotesovec, Jan 13 2014
EXAMPLE
E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 231*x^5/5! + 2701*x^6/6! +...
MATHEMATICA
Rest[CoefficientList[InverseSeries[Series[Log[1+x]*Sqrt[1-x^2], {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 13 2014 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(serreverse(log(1+X)*sqrt(1-X^2)), n)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 20 2013
STATUS
approved