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%I #4 Dec 24 2012 11:46:15
%S 1,1,4,20,111,657,4067,26028,170913,1145446,7804797,53911104,
%T 376669462,2657391772,18904566514,135460704648,976795422828,
%U 7082951967141,51614974500605,377798933519164,2776363089297553,20476554379564305
%N G.f.: A(x) = F(x*G(x)) 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-k,n-k)*k/(2n-k) for n>0 with a(0)=1.
%F G.f. satisfies: A(x) = 1 + x*G(x)*A(x)^3 where G(x) is the g.f. of A000108.
%F G.f. satisfies: A(x*F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A001764.
%e G.f.: A(x) = F(x*G(x)) = 1 + x + 4*x^2 + 20*x^3 + 111*x^4 +...
%e Related expansions.
%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 +...
%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 + 9*x^2 + 48*x^3 + 278*x^4 + 1696*x^5 +...
%e A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 513*x^4 + 3225*x^5 +...
%e G(x)*A(x)^3 = 1 + 4*x + 20*x^2 + 111*x^3 + 657*x^4 +...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(2*(n-k)+k,n-k)*k/(2*(n-k)+k)))}
%Y Cf. A000108, A001764; A153298, A153390.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 15 2009
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