A153297 G.f.: A(x) = F(x*G(x)^2)^2 where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 2, 9, 48, 276, 1656, 10212, 64190, 409218, 2637282, 17143506, 112224228, 738926064, 4889332266, 32488240779, 216664589058, 1449568426292, 9725637277248, 65417353098837, 441013558347228, 2979206654245122

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} C(2k+2,k)/(k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.

G.f. satisfies: A(x/F(x)) = F(x*F(x))^2 where F(x) is the g.f. of A000108 (Catalan).

Example

G.f.: A(x) = F(x*G(x)^2)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 276*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(2*k+2, k)*2/(2*k+2)*binomial(3*(n-k)+2*k, n-k)*2*k/(3*(n-k)+2*k)))}

Crossrefs

Cf. A000108, A001764; A153296, A153298.

Sequence in context: A190253 A174687 A047059 * A364734 A153390 A118341

Adjacent sequences: A153294 A153295 A153296 * A153298 A153299 A153300

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved