A153398 G.f.: A(x) = F(x*G(x)^2) 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, 5, 33, 245, 1941, 16023, 136075, 1179833, 10392981, 92701411, 835271032, 7589337123, 69444928453, 639280878401, 5915683250220, 54991636090761, 513257729193329, 4807619948647095, 45177320023095160, 425766248463523359

Offset

0,3

Links

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

Formula

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

G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^3 where G(x) is the g.f. of A002293.

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

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

Example

G.f.: A(x) = F(x*G(x)^2) = 1 + x + 5*x^2 + 33*x^3 + 245*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 + 11*x^2 + 76*x^3 + 581*x^4 + 4702*x^5 +...
A(x)^3 = 1 + 3*x + 18*x^2 + 130*x^3 + 1023*x^4 + 8457*x^5 +...
G(x)^2*A(x)^3 = 1 + 5*x + 33*x^2 + 245*x^3 + 1941*x^4 + 16023*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)+2*k, n-k)*2*k/(4*(n-k)+2*k)))}

Crossrefs

Cf. A000108, A001764, A002293; A153397, A153399.

Sequence in context: A142989 A084131 A084771 * A242522 A034015 A268563

Adjacent sequences: A153395 A153396 A153397 * A153399 A153400 A153401

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved