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

0,3

Links

Table of n, a(n) for n=0..19.

Formula

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.

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

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

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.

Example

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 +...

Prog

(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)))}

Crossrefs

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

Sequence in context: A257633 A371779 A007193 * A007194 A025551 A101600

Adjacent sequences: A153396 A153397 A153398 * A153400 A153401 A153402

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved