A153391 G.f.: A(x) = F(x*G(x)^2) 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, 5, 29, 183, 1223, 8525, 61366, 453003, 3412077, 26124599, 202748728, 1591450129, 12612760009, 100790253764, 811227147197, 6570431009209, 53512143110041, 437976298197769, 3600504527707557, 29716593448484673

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,n-k)*k/n for n>0 with a(0)=1.

G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A000108.

G.f. satisfies: A(x*F(x)) = F(F(x)-1) where F(x) is the g.f. of A001764.

Example

G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 183*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 +...
A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 449*x^4 + 3102*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*x^4 + 5799*x^5 +...
G(x)^2*A(x)^3 = 1 + 5*x + 29*x^2 + 183*x^3 + 1223*x^4 + 8525*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)+2*k, n-k)*2*k/(2*(n-k)+2*k)))}

Crossrefs

Cf. A000108, A001764; A153390, A153392.

Sequence in context: A153296 A194723 A190917 * A175891 A081336 A127846

Adjacent sequences: A153388 A153389 A153390 * A153392 A153393 A153394

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved