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

0,3

Links

Table of n, a(n) for n=0..21.

Formula

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

G.f. satisfies: A(x) = 1 + x*G(x)*A(x)^3 where G(x) is the g.f. of A000108.

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

Example

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 +...

Prog

(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)))}

Crossrefs

Cf. A000108, A001764; A153298, A153390.

Sequence in context: A026156 A025183 A014523 * A239643 A081335 A136783

Adjacent sequences: A153296 A153297 A153298 * A153300 A153301 A153302

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved