OFFSET
0,3
COMMENTS
Compare to the identity: 1/(1-x) = Sum_{n>=0} n!*x^n/Product_{k=1..n} (1+k*x).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 38*x^4 + 302*x^5 + 3428*x^6 +...
where, by definition,
A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)*A(2*x)/((1+x*A(x))*(1+2*x*A(2*x))) + 3!*x^3*A(x)*A(2*x)*A(3*x)/((1+x*A(x))*(1+2*x*A(2*x))*(1+3*x*A(3*x))) + 4!*x^4*A(x)*A(2*x)*A(3*x)*A(4*x)/((1+x*A(x))*(1+2*x*A(2*x))*(1+3*x*A(3*x))*(1+4*x*A(4*x))) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*prod(k=1, m, subst(A, x, k*x+x*O(x^n ))/(1+k*x*subst(A, x, k*x+x*O(x^n)))))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 06 2013
STATUS
approved