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A153298 G.f.: A(x) = F(x*G(x)^3)^2 = F(G(x)-1)^2 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, 2, 11, 68, 443, 2974, 20361, 141356, 991738, 7015814, 49967892, 357896120, 2575844046, 18616823352, 135051785186, 982949932092, 7175591019313, 52524480778590, 385429134781530, 2834791998208500, 20893844524709649 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

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

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

EXAMPLE

G.f.: A(x) = F(x*G(x)^3)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*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 + 7752*x^6 +...

PROG

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

CROSSREFS

Cf. A000108, A001764; A153297, A153299.

Sequence in context: A266579 A063768 A042245 * A153393 A274736 A229230

Adjacent sequences:  A153295 A153296 A153297 * A153299 A153300 A153301

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 15 2009

STATUS

approved

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)