OFFSET
0,2
COMMENTS
More generally, we have the identity:
Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=A(x).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..100
FORMULA
E.g.f. satisfies: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + k*A(x)).
a(n) ~ s*sqrt(r*(1+s)*(r*s-1)/(1-4*r*s+r^2*s*(2*s-1))) * n^(n-1) / (exp(n) * r^n), where s = 2.4590533113276368838... is the root of the equation (1+s)*(1+2*s) = s^(s/(1+s))*(1+s)^2 - s*log(s) and r = (1 - s^(-s/(1+s)))/s = 0.1921573821382919835... - Vaclav Kotesovec, Jan 11 2014
EXAMPLE
E.g.f.: A(x) = 1 + 2*x + 10*x^2/2! + 96*x^3/3! + 1388*x^4/4! + 26960*x^5/5! +...
where e.g.f. A = A(x) satisfies:
A = 1 + x*(1+A) + x^2*(1+A)*(1+2*A)/2! + x^3*(1+A)*(1+2*A)*(1+3*A)/3! + x^4*(1+A)*(1+2*A)*(1+3*A)*(1+4*A)/4! + x^5*(1+A)*(1+2*A)*(1+3*A)*(1+4*A)*(1+5*A)/5! +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1/(1-x*A +x*O(x^n))^(1+1/A)); n!*polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/m!*prod(k=1, m, 1+k*A+x*O(x^n)))); n!*polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 21 2011
STATUS
approved