OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..402
FORMULA
E.g.f.: A(x) = 1 - LambertW(-x^2*exp(x))/x.
E.g.f.: A(x) = 1 + Sum_{n>=1} x^(2*n-1) * n^(n-1) * exp(n*x) / n!.
E.g.f.: A(x) = (1/x)*Series_Reversion(x/B(x)) where B(x) = 1 + x*exp(x)/B(x) = (1+sqrt(1+4*x*exp(x)))/2.
a(n) = n!*Sum_{k=0..n} C(n-k+1,k)/(n-k+1) * k^(n-k)/(n-k)!.
If A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
a(n,m) = n!*Sum_{k=0..n} m*C(n-k+m,k)/(n-k+m) * k^(n-k)/(n-k)!.
a(n) ~ sqrt(1+LambertW(1/(2*exp(1/2)))) * n^(n-1) / (exp(n) * 2^(n+1/2) * (LambertW(1/(2*exp(1/2))))^(n+1)). - Vaclav Kotesovec, Jul 09 2013
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 52*x^4/4! + 425*x^5/5! +...
exp(x*A(x)) = 1 + x + 3*x^2/2! + 13*x^3/3! + 85*x^4/4! + 701*x^5/5! +...
MATHEMATICA
CoefficientList[Series[1-LambertW[-x^2*E^x]/x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jul 09 2013 *)
PROG
(PARI) {a(n, m=1)=n!*sum(k=0, n, m*binomial(n-k+m, k)/(n-k+m)*k^(n-k)/(n-k)!)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 18 2009
STATUS
approved