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A153296 G.f.: A(x) = F(x*G(x)^3) = F(G(x)-1) where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764. 2
1, 1, 5, 29, 180, 1162, 7698, 51950, 355531, 2460224, 17178755, 120861710, 855828960, 6094211829, 43610311298, 313449094851, 2261820356684, 16379528485200, 119003715014955, 867198605427231, 6336861345197670 (list; graph; refs; listen; history; text; internal format)
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(3n,n-k)*k/n for n>0 with a(0)=1.

G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^2 where G(x) is the g.f. of A001764.

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

EXAMPLE

G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 29*x^3 + 180*x^4 +... where

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 + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...

G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...

G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...

A(x)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 + 2974*x^5 +...

G(x)^3*A(x)^2 = 1 + 5*x + 29*x^2 + 180*x^3 + 1162*x^4 +...

PROG

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

CROSSREFS

Cf. A000108, A001764; A153295, A153297.

Sequence in context: A190802 A139174 A290117 * A194723 A190917 A153391

Adjacent sequences:  A153293 A153294 A153295 * A153297 A153298 A153299

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 15 2009

STATUS

approved

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Last modified February 23 09:04 EST 2019. Contains 320420 sequences. (Running on oeis4.)