A153397 G.f.: A(x) = F(x*G(x)^4) = F(G(x)-1) where F(x) = G(x/F(x)^2) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)^2) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 6, 43, 334, 2717, 22776, 195000, 1695874, 14927990, 132673398, 1188412986, 10714602196, 97133633788, 884716464592, 8091061578807, 74259516900390, 683694381314696, 6312247839166260, 58424001667319720, 541971167468786770

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(4n,n-k)*k/n for n>0 with a(0)=1.

G.f. satisfies: A(x) = 1 + x*G(x)^4*A(x)^2 where G(x) is the g.f. of A002293.

G.f. satisfies: A(x/F(x)^2) = F(F(x)-1) where F(x) is the g.f. of A000108.

G.f. satisfies: A(x/H(x)) = F(H(x)-1) where H(x) = 1 + x*H(x)^3 is the g.f. of A001764 and F(x) is the g.f. of A000108.

Example

G.f.: A(x) = F(x*G(x)^4) = 1 + x + 6*x^2 + 43*x^3 + 334*x^4 +...
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*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 + 98*x^3 + 790*x^4 + 6618*x^5 +...
G(x)^4*A(x)^2 = 1 + 6*x + 43*x^2 + 334*x^3 + 2717*x^4 + 22776*x^5 +...

Prog

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*binomial(4*(n-k)+4*k, n-k)*4*k/(4*(n-k)+4*k)))}

Crossrefs

Cf. A000108, A001764, A002293; A153396, A153398.

Sequence in context: A349302 A025594 A098665 * A005786 A071541 A146966

Adjacent sequences: A153394 A153395 A153396 * A153398 A153399 A153400

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved