A153391 - OEIS

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

%S 1,1,5,29,183,1223,8525,61366,453003,3412077,26124599,202748728,

%T 1591450129,12612760009,100790253764,811227147197,6570431009209,

%U 53512143110041,437976298197769,3600504527707557,29716593448484673

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

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

%F G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A000108.

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

%e G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 183*x^4 +... where

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

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

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

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

%e A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 449*x^4 + 3102*x^5 +...

%e A(x)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*x^4 + 5799*x^5 +...

%e G(x)^2*A(x)^3 = 1 + 5*x + 29*x^2 + 183*x^3 + 1223*x^4 + 8525*x^5 +...

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

%Y Cf. A000108, A001764; A153390, A153392.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 15 2009