A226761 G.f.: 1 / (1 + 12*x*G(x)^2 - 13*x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

1, 1, 16, 118, 1004, 8601, 75076, 662796, 5903676, 52949332, 477533356, 4326309406, 39343725716, 358943047438, 3283745710968, 30112624408488, 276715616909148, 2547523969430508, 23491659440021920, 216942761366305144, 2006084011596742384, 18572529488934397689

Offset: 0

Author

Paul D. Hanna, Jun 16 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 16*x^2 + 118*x^3 + 1004*x^4 + 8601*x^5 +...
A related series is G(x) = 1 + x*G(x)^4, where
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 + 17710*x^6 +...
G(x)^3 = 1 + 3*x + 15*x^2 + 91*x^3 + 612*x^4 + 4389*x^5 + 32890*x^6 +...
such that A(x) = 1/(1 + 12*x*G(x)^2 - 13*x*G(x)^3).
		

Links

• Vincenzo Librandi and Joerg Arndt, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A002293.

Cf. A005810, A078995, A147855, A226733, A385605, A386811.

Programs

Mathematica

Table[Sum[Binomial[2*n+3*k,n-k]*Binomial[2*n-3*k,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jun 17 2013 *)

PARI

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1+12*x*G^2-13*x*G^3), n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=local(G=1+x); for(i=0, n, G=1+x*G^4+x*O(x^n)); polcoeff(1/(1-x*G^2-13*x^2*G^6), n)}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=sum(k=0, n, binomial(2*n+3*k, n-k)*binomial(2*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=sum(k=0, n, binomial(3*k, n-k)*binomial(4*n-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

PARI

{a(n)=sum(k=0, n, binomial(4*n+3*k, n-k)*binomial(-3*k, k))}
for(n=0, 30, print1(a(n), ", "))

Formula

a(n) = Sum_{k=0..n} C(3*k, n-k) * C(4*n-3*k, k).

a(n) = Sum_{k=0..n} C(n+3*k, n-k) * C(3*n-3*k, k).

a(n) = Sum_{k=0..n} C(2*n+3*k, n-k) * C(2*n-3*k, k).

a(n) = Sum_{k=0..n} C(3*n+3*k, n-k) * C(n-3*k, k).

a(n) = Sum_{k=0..n} C(4*n+3*k, n-k) * C(-3*k, k).

G.f.: 1 / (1 - x*G(x)^2 - 13*x^2*G(x)^6) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

a(n) ~ 2^(8*n+5/2)/(7*3^(3*n+1/2)*sqrt(Pi*n)). - Vaclav Kotesovec, Jun 17 2013

From Seiichi Manyama, Aug 05 2025: (Start)

a(n) = [x^n] 1/((1+3*x) * (1-x)^(3*n+1)).

a(n) = Sum_{k=0..n} (-4)^(n-k) * binomial(4*n+1,k).

a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n+k,k). (End)

From Seiichi Manyama, Aug 14 2025: (Start)

a(n) = Sum_{k=0..n} (-3)^k * 4^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

G.f.: G(x)^2/((-3+4*G(x)) * (4-3*G(x))) where G(x) = 1+x*G(x)^4 is the g.f. of A002293. (End)

G.f.: B(x)^2/(1 + 7*(B(x)-1)/4), where B(x) is the g.f. of A005810. - Seiichi Manyama, Aug 15 2025