OFFSET
0,3
FORMULA
E.g.f. satisfies: A(x) = exp(-1) * Sum_{n>=0} exp(x*A(x)^(n+1))/n!.
E.g.f. satisfies: A(x) = G(x*A(x)) where G(x) = A(x/G(x)) is the e.g.f. of A195895; thus, A(x) = (1/x)*Series_Reversion(x/G(x)).
EXAMPLE
E.g.f.: A(x) = 1 + x + 5*x^2/2! + 52*x^3/3! + 853*x^4/4! + 19291*x^5/5! +...
where
A(x) = 1 + exp(A(x)-1)*A(x)*x + exp(A(x)^2-1)*A(x)^2*x^2/2! + exp(A(x)^3-1)*A(x)^3*x^3/3! +...
Also, A(x) = G(x*A(x)) where G(x) = A(x/G(x)) = e.g.f. of A195895(n):
G(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 2996*x^5/5! + 57613*x^6/6! +...+ A195895(n)*x^n/n! +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, exp(A^m-1+x*O(x^n))*A^m*x^m/m!)); n!*polcoeff(A, n)}
(PARI) /* Alternate e.g.f. (requires high precision): */
{a(n)=local(A=1+x); for(i=1, n, A=exp(-1)*sum(m=0, 2*n+10, exp(x*A^(m+1)+x*O(x^n))/m!)); round(n!*polcoeff(A, n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 03 2011
STATUS
approved