OFFSET
0,3
COMMENTS
In general, if e.g.f. satisfies: A(x) = exp( Integral(1 + x*A(x)^p) dx ), p>1, and the constant of integration is zero, then A(x) = (1/p + (p-1)/(exp(p*x)*p) - x)^(-1/p), and a(n) ~ n! * p^(n+1/p) / (Gamma(1/p) * n^(1-1/p) * (1+LambertW((p-1)*exp(-1)))^(n+2/p)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..380
FORMULA
E.g.f.: 5^(1/5)*exp(x)/(exp(5*x) - 5*exp(5*x)*x + 4)^(1/5).
a(n) ~ Gamma(4/5) * sqrt(5-sqrt(5)) * 5^(n+1/5) * n^(n-3/10) / (2*sqrt(Pi) * exp(n) * (LambertW(4*exp(-1))+1)^(n+2/5)). - Vaclav Kotesovec, Jul 15 2014
MATHEMATICA
CoefficientList[Series[(1/p + (p-1)/(E^(p*x)*p) - x)^(-1/p) /. p->5, {x, 0, 20}], x] * Range[0, 20]!
PROG
(PARI) x='x+O('x^30); Vec(serlaplace(round(5^(1/5)*exp(x)/(exp(5*x) - 5*exp(5*x)*x + 4)^(1/5)))) \\ G. C. Greubel, Sep 09 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Vaclav Kotesovec, Jul 15 2014
STATUS
approved