A153296 - OEIS

%I #2 Mar 30 2012 18:37:15

%S 1,1,5,29,180,1162,7698,51950,355531,2460224,17178755,120861710,

%T 855828960,6094211829,43610311298,313449094851,2261820356684,

%U 16379528485200,119003715014955,867198605427231,6336861345197670

%N 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.

%F 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.

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

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

%e G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 180*x^4 +... where

%e F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...

%e F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...

%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

%e G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...

%e G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...

%e A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 + 2974*x^5 +...

%e G(x)^3*A(x)^2 = 1 + 5*x + 29*x^2 + 180*x^3 + 1162*x^4 +...

%o (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)))}

%Y Cf. A000108, A001764; A153295, A153297.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 15 2009