OFFSET
0,3
COMMENTS
Compare to the g.f. C(x) = 1 + x*C(x)^2 of the Catalan numbers (A000108): C(x) = exp( Sum_{n>=1} C(2*n,n)/2 * x^n/n ).
FORMULA
G.f.: Product_{n>=1} C(x^n)^a(n-1) = Sum_{n>=0} a(n)*x^n, where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 97*x^5 + 336*x^6 +...
where
log(A(x)) = A(x)*x + 3*A(x^2)*x^2/2 + 10*A(x^3)*x^3/3 + 35*A(x^4)*x^4/4 + 126*A(x^5)*x^5/5 + 462*A(x^6)*x^6/6 + 1716*A(x^7)*x^7/7 + 6435*A(x^8)*x^8/8 +...
The g.f. also equals the product:
A(x) = C(x) * C(x^2) * C(x^3)^3 * C(x^4)^9 * C(x^5)^30 * C(x^6)^97 *...* C(x^n)^a(n-1) *...
where C(x) is the g.f. of the Catalan numbers:
C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, subst(A, x, x^m+x*O(x^n))*binomial(2*m, m)/2*x^m/m))); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 03 2011
STATUS
approved