A153298 G.f.: A(x) = F(x*G(x)^3)^2 = F(G(x)-1)^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, 11, 68, 443, 2974, 20361, 141356, 991738, 7015814, 49967892, 357896120, 2575844046, 18616823352, 135051785186, 982949932092, 7175591019313, 52524480778590, 385429134781530, 2834791998208500, 20893844524709649

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,n-k)*k/n for n>0 with a(0)=1.

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

Example

G.f.: A(x) = F(x*G(x)^3)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*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)+3*k, n-k)*3*k/(3*(n-k)+3*k)))}

Crossrefs

Cf. A000108, A001764; A153297, A153299.

Sequence in context: A266579 A063768 A042245 * A153393 A365135 A274736

Adjacent sequences: A153295 A153296 A153297 * A153299 A153300 A153301

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved