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G.f.: A(x) = F(x*G(x)^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)^4 is the g.f. of A002293.
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%I #4 Mar 23 2022 09:42:32

%S 1,1,6,45,371,3225,29007,267239,2506605,23842644,229369064,2227345899,

%T 21801617643,214862158025,2130226863222,21231722675274,

%U 212613977684254,2138164077605865,21585420400120710,218677042735538547

%N G.f.: A(x) = F(x*G(x)^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)^4 is the g.f. of A002293.

%F a(n) = Sum_{k=0..n} C(3*k+1,k)/(3*k+1) * C(4*n-k,n-k)*3*k/(4*n-k) for n>0 with a(0)=1.

%F G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^3 where G(x) is the g.f. of A002293.

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

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

%e G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 45*x^3 + 371*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 + 7752*x^6 +...

%e G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +...

%e G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +...

%e G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +...

%e G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +...

%e A(x)^2 = 1 + 2*x + 13*x^2 + 102*x^3 + 868*x^4 + 7732*x^5 +...

%e A(x)^3 = 1 + 3*x + 21*x^2 + 172*x^3 + 1509*x^4 + 13764*x^5 +...

%e G(x)^3*A(x)^3 = 1 + 6*x + 45*x^2 + 371*x^3 + 3225*x^4 + 29007*x^5 +...

%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+1,k)/(3*k+1)*binomial(4*(n-k)+3*k,n-k)*3*k/(4*(n-k)+3*k)))}

%Y Cf. A000108, A001764, A002293; A153398, A153291.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 15 2009