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

0,3

Links

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

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

Example

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).

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A147855, A226733, A002293.

Sequence in context: A302897 A145216 A200173 * A250169 A210428 A283037

Adjacent sequences: A226758 A226759 A226760 * A226762 A226763 A226764

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2013

Status

approved