A226751 G.f.: 1 / (1 + 6*x*G(x) - 7*x*G(x)^2), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

1, 1, 9, 48, 289, 1761, 10932, 68664, 435201, 2777763, 17829489, 114968052, 744178716, 4832624044, 31469746632, 205422018288, 1343734578561, 8806130111847, 57805893969531, 380013533789928, 2501507255441049, 16486378106441697, 108773240389894056

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(2*k, n-k) * C(3*n-2*k, k).

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

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

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

G.f.: 1/(1 - x*G(x) - 7*x^2*G(x)^4), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

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

Conjecture: 18*n*(2*n-1)*(55*n-76)*a(n) +(-11605*n^3+28521*n^2-20870*n+4536)*a(n-1) -24*(55*n-21)*(3*n-4)*(3*n-2)*a(n-2)=0. - R. J. Mathar, Jun 14 2016

Example

G.f.: A(x) = 1 + x + 9*x^2 + 48*x^3 + 289*x^4 + 1761*x^5 + 10932*x^6 +...
A related series is G(x) = 1 + x*G(x)^3, where
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 +...
such that A(x) = 1/(1 + 6*x*G(x) - 7*x*G(x)^2).

Mathematica

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

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=sum(k=0, n, binomial(2*k, n-k)*binomial(3*n-2*k, k))}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^3+x*O(x^n)); polcoeff(1/(1+6*x*G-7*x*G^2), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(G=1+x); for(i=0, n, G=1+x*G^3+x*O(x^n)); polcoeff(1/(1-x*G-7*x^2*G^4), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A183160, A147855, A001764.

Sequence in context: A181959 A171011 A264273 * A073584 A007037 A179280

Adjacent sequences: A226748 A226749 A226750 * A226752 A226753 A226754

Keyword

nonn

Author

Paul D. Hanna, Jun 16 2013

Status

approved