A153392 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)^2 is the g.f. of A000108 (Catalan).

1, 1, 6, 39, 272, 2001, 15333, 121266, 983274, 8133564, 68382628, 582700485, 5021538753, 43690059657, 383263396836, 3386175566418, 30104702903914, 269125162789764, 2417709649413102, 21815252320257250, 197620659225838530

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

a(n) = Sum_{k=0..n} C(3k+1,k)/(3k+1) * C(2n+k,n-k)*3k/(2n+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 A000108.

G.f. satisfies: A(x*F(x)) = F(x*F(x)^4) where F(x) is the g.f. of A001764.

Example

G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 39*x^3 + 272*x^4+... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
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 + 429*x^6 +...
G(x)^3 = 1 + 3*x + 9*x^2 + 28*x^3 + 90*x^4 + 297*x^5 + 1001*x^6 +...
A(x)^2 = 1 + 2*x + 13*x^2 + 90*x^3 + 658*x^4 + 5014*x^5 +...
A(x)^3 = 1 + 3*x + 21*x^2 + 154*x^3 + 1176*x^4 + 9264*x^5 +...
G(x)^3*A(x)^3 = 1 + 6*x + 39*x^2 + 272*x^3 + 2001*x^4 + 15333*x^5 +...

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*binomial(2*(n-k)+3*k, n-k)*3*k/(2*(n-k)+3*k)))}

Crossrefs

Cf. A000108, A001764; A153391, A153393.

Sequence in context: A068765 A349531 A006633 * A253077 A231482 A367233

Adjacent sequences: A153389 A153390 A153391 * A153393 A153394 A153395

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved