A153394 G.f.: A(x) = F(x*G(x)^2)^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, 3, 18, 118, 813, 5799, 42470, 317637, 2416671, 18649874, 145655292, 1149199212, 9146686605, 73354982763, 592217363334, 4809250320023, 39258457746069, 321964620209940, 2651536017682988, 21919266484180533, 181820251665093357

Offset

0,2

Links

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

Formula

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

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

Example

G.f.: A(x) = F(x*G(x)^2)^3 = 1 + 3*x + 18*x^2 + 118*x^3 + 813*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 +...
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 +...

Prog

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

Crossrefs

Cf. A000108, A001764; A153393, A153395.

Sequence in context: A291076 A113328 A196865 * A334682 A320616 A009065

Adjacent sequences: A153391 A153392 A153393 * A153395 A153396 A153397

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved