A153299 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).
1, 1, 4, 20, 111, 657, 4067, 26028, 170913, 1145446, 7804797, 53911104, 376669462, 2657391772, 18904566514, 135460704648, 976795422828, 7082951967141, 51614974500605, 377798933519164, 2776363089297553, 20476554379564305
Offset: 0
Keywords
Examples
G.f.: A(x) = F(x*G(x)) = 1 + x + 4*x^2 + 20*x^3 + 111*x^4 +... Related expansions. F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +... F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +... G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +... G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 +... A(x)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 278*x^4 + 1696*x^5 +... A(x)^3 = 1 + 3*x + 15*x^2 + 85*x^3 + 513*x^4 + 3225*x^5 +... G(x)*A(x)^3 = 1 + 4*x + 20*x^2 + 111*x^3 + 657*x^4 +...
Programs
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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)))}