OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..100
FORMULA
E.g.f. satisfies: A(x) = 1 + x/(1 + LambertW(-log(A(x)))).
a(n) ~ n^(n-1) * sqrt(s*(s-1)^3/((r-1)*s+2*s^2-1)) / (exp(n) * r^n), where s = 1.30989890437082404330133094063455155... is the root of the equation 1 + (s-1)*LambertW(-log(s)) / (s*log(s)*(1 + LambertW(-log(s)))^2) = 0, and r = (s-1)*(1+LambertW(-log(s))) = 0.1845269281080403527171896528382422... - Vaclav Kotesovec, Feb 28 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 2*x^2/2! + 15*x^3/3! + 200*x^4/4! + 3760*x^5/5! +...
where
A(x) = 1 + x + x*log(A(x)) + 2^2*x*log(A(x))^2/2! + 3^3*x*log(A(x))^3/3! + 4^4*x*log(A(x))^4/4! + 5^5*x*log(A(x))^5/5! +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*sum(m=0, n, log( subst(A^m, x, x+x*O(x^n)) )^m/m!)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 01 2013
STATUS
approved