A153399 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.
1, 1, 6, 45, 371, 3225, 29007, 267239, 2506605, 23842644, 229369064, 2227345899, 21801617643, 214862158025, 2130226863222, 21231722675274, 212613977684254, 2138164077605865, 21585420400120710, 218677042735538547
Offset: 0
Keywords
Examples
G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 45*x^3 + 371*x^4 +... where 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 + 3876*x^6 +... F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +... G(x) = 1 + x + 4*x^2 + 22*x^3 + 140*x^4 + 969*x^5 + 7084*x^6 +... G(x)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 340*x^4 + 2394*x^5 +... G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 +... G(x)^4 = 1 + 4*x + 22*x^2 + 140*x^3 + 969*x^4 + 7084*x^5 +... A(x)^2 = 1 + 2*x + 13*x^2 + 102*x^3 + 868*x^4 + 7732*x^5 +... A(x)^3 = 1 + 3*x + 21*x^2 + 172*x^3 + 1509*x^4 + 13764*x^5 +... G(x)^3*A(x)^3 = 1 + 6*x + 45*x^2 + 371*x^3 + 3225*x^4 + 29007*x^5 +...
Programs
-
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)))}