A153296 G.f.: A(x) = F(x*G(x)^3) = F(G(x)-1) 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, 1, 5, 29, 180, 1162, 7698, 51950, 355531, 2460224, 17178755, 120861710, 855828960, 6094211829, 43610311298, 313449094851, 2261820356684, 16379528485200, 119003715014955, 867198605427231, 6336861345197670

Offset: 0

Author

Paul D. Hanna, Jan 15 2009

Keywords

• nonn

Examples

			G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 180*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 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 + 2974*x^5 +...
G(x)^3*A(x)^2 = 1 + 5*x + 29*x^2 + 180*x^3 + 1162*x^4 +...
		

Crossrefs

Cf. A000108, A001764; A153295, A153297.

Programs

PARI

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

Formula

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

G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^2 where G(x) is the g.f. of A001764.

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