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%I #2 Mar 30 2012 18:37:15
%S 1,1,6,43,334,2717,22776,195000,1695874,14927990,132673398,1188412986,
%T 10714602196,97133633788,884716464592,8091061578807,74259516900390,
%U 683694381314696,6312247839166260,58424001667319720,541971167468786770
%N G.f.: A(x) = F(x*G(x)^4) = F(G(x)-1) where F(x) = G(x/F(x)^2) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)^2) = 1 + x*G(x)^4 is the g.f. of A002293.
%F a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(4n,n-k)*k/n for n>0 with a(0)=1.
%F G.f. satisfies: A(x) = 1 + x*G(x)^4*A(x)^2 where G(x) is the g.f. of A002293.
%F G.f. satisfies: A(x/F(x)^2) = F(F(x)-1) where F(x) is the g.f. of A000108.
%F G.f. satisfies: A(x/H(x)) = F(H(x)-1) where H(x) = 1 + x*H(x)^3 is the g.f. of A001764 and F(x) is the g.f. of A000108.
%e G.f.: A(x) = F(x*G(x)^4) = 1 + x + 6*x^2 + 43*x^3 + 334*x^4 +...
%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 + 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 + 98*x^3 + 790*x^4 + 6618*x^5 +...
%e G(x)^4*A(x)^2 = 1 + 6*x + 43*x^2 + 334*x^3 + 2717*x^4 + 22776*x^5 +...
%o (PARI) {a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*binomial(4*(n-k)+4*k,n-k)*4*k/(4*(n-k)+4*k)))}
%Y Cf. A000108, A001764, A002293; A153396, A153398.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 15 2009
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