OFFSET
0,2
COMMENTS
A variant is A007857, the number of independent sets in rooted plane trees on n nodes.
FORMULA
EXAMPLE
G.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 397*x^4 + 2802*x^5 +...;
A(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
PROG
(PARI) {a(n)=local(m=n+1, C, F, A); C=Ser(vector(m, r, binomial(2*r-2, r-1)/r)); F=Ser(vector(m, r, binomial(4*r-4, r-1)/(3*r-2))); A=1/(1-x*C*F^2-x*F^3); polcoeff(A+O(x^m), n, x)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 27 2008
STATUS
approved