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A153399 G.f.: A(x) = F(x*G(x)^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)^4 is the g.f. of A002293. 1
1, 1, 6, 45, 371, 3225, 29007, 267239, 2506605, 23842644, 229369064, 2227345899, 21801617643, 214862158025, 2130226863222, 21231722675274, 212613977684254, 2138164077605865, 21585420400120710, 218677042735538547 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(4n-k,n-k)*3k/(4n-k) for n>0 with a(0)=1.

G.f. satisfies: A(x) = 1 + x*G(x)^3*A(x)^3 where G(x) is the g.f. of A002293.

G.f. satisfies: A(x/F(x)) = F(x*F(x)^2) where F(x) is the g.f. of A001764.

G.f. satisfies: A(x/H(x)^2) = F(x*H(x)) where H(x) = 1 + x*H(x)^2 is the g.f. of A000108 (Catalan) and F(x) is the g.f. of A001764.

EXAMPLE

G.f.: A(x) = F(x*G(x)^3) = 1 + x + 6*x^2 + 45*x^3 + 371*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 + 7752*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 + 102*x^3 + 868*x^4 + 7732*x^5 +...

A(x)^3 = 1 + 3*x + 21*x^2 + 172*x^3 + 1509*x^4 + 13764*x^5 +...

G(x)^3*A(x)^3 = 1 + 6*x + 45*x^2 + 371*x^3 + 3225*x^4 + 29007*x^5 +...

PROG

(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(3*k+1, k)/(3*k+1)*binomial(4*(n-k)+3*k, n-k)*3*k/(4*(n-k)+3*k)))}

CROSSREFS

Cf. A000108, A001764, A002293; A153398, A153291.

Sequence in context: A004988 A257633 A007193 * A007194 A025551 A101600

Adjacent sequences:  A153396 A153397 A153398 * A153400 A153401 A153402

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 15 2009

STATUS

approved

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Last modified November 19 00:12 EST 2019. Contains 329310 sequences. (Running on oeis4.)