A153394 - OEIS

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

%S 1,3,18,118,813,5799,42470,317637,2416671,18649874,145655292,

%T 1149199212,9146686605,73354982763,592217363334,4809250320023,

%U 39258457746069,321964620209940,2651536017682988,21919266484180533,181820251665093357

%N G.f.: A(x) = F(x*G(x)^2)^3 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+3,k)/(k+1) * C(2n,n-k)*k/n for n>0 with a(0)=1.

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

%e G.f.: A(x) = F(x*G(x)^2)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*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)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*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 + 429*x^6 +...

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

%Y Cf. A000108, A001764; A153393, A153395.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 15 2009