OFFSET
0,3
COMMENTS
Compare the g.f. to the identities:
(1) 1/(1-x) = Sum_{n>=0} n!*x^n / Product_{k=1..n} (1 + k*x).
(2) C(x) = Sum_{n>=0} n!*x^n*C(x)^n / Product_{k=1..n} (1 + k*x*C(x)), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 54*x^4 + 392*x^5 + 3264*x^6 +...
where
A(x) = 1 + x*A(x)/(1+x*A(x)) + 2!*x^2*A(x)^4/((1+x*A(x)^2)*(1+2*x*A(x)^2)) + 3!*x^3*A(x)^9/((1+x*A(x)^3)*(1+2*x*A(x)^3)*(1+3*x*A(x)^3)) + 4!*x^4*A(x)^16/((1+x*A(x)^4)*(1+2*x*A(x)^4)*(1+3*x*A(x)^4)*(1+4*x*A(x)^4)) +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, m!*x^m*A^(m^2)/prod(k=1, m, 1+k*x*(A+x*O(x^n))^m))); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 04 2013
STATUS
approved