A137571 Main diagonal of square array A137570.

1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, 81394123, 672998498, 5628741195, 47535483498, 404790717079, 3471892750622, 29966295451511, 260080708564964, 2268416956569463, 19872441881999354, 174783803353387498

Offset

0,2

Comments

A variant is A007857, the number of independent sets in rooted plane trees on n nodes.

Links

Table of n, a(n) for n=0..20.

Formula

G.f. 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) and F(x) = 1 + xF(x)^4 is g.f. of A002293.

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

Cf. A137570, A137572, A137573; A007857 (variant); A000108, A002293.

Sequence in context: A004981 A214764 A365554 * A215002 A301625 A262001

Adjacent sequences: A137568 A137569 A137570 * A137572 A137573 A137574

Keyword

nonn

Author

Paul D. Hanna, Jan 27 2008

Status

approved