A188911 Binomial convolution of the binomial coefficients bin(3n,n) (A005809).

1, 6, 48, 438, 4356, 46056, 509106, 5814738, 68050116, 811240872, 9810384048, 119990105208, 1481115683754, 18421300391760, 230574816629310, 2901721280735838, 36688485233689668, 465774244616805624, 5934465567864915024

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..90

Formula

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

E.g.f.: F(1/3,2/3;1/2,1;27*x/4)^2, where F(a1,a2;b1,b2;z) is a hypergeometric series.

Recurrence: 8*n^2 * (2*n-1)^2 * (9*n^3 - 54*n^2 + 102*n - 61)*a(n) = 24*(3*n-1)*(108*n^6 - 855*n^5 + 2628*n^4 - 4059*n^3 + 3380*n^2 - 1470*n + 264)*a(n-1) - 18*(3645*n^7 - 34992*n^6 + 138348*n^5 - 291843*n^4 + 352980*n^3 - 241794*n^2 + 84684*n - 11104)*a(n-2) + 2187*(n-2)^2 * (3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 21*n - 4)*a(n-3). - Vaclav Kotesovec, Feb 25 2014

a(n) ~ 3^(3*n+1) / (Pi * n * 2^(n+1)). - Vaclav Kotesovec, Feb 25 2014

Mathematica

Table[Sum[Binomial[n, k]Binomial[3k, k]Binomial[3n-3k, n-k], {k, 0, n}], {n, 0, 22}]

Prog

(Maxima) makelist(sum(binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k), k, 0, n), n, 0, 12);
(PARI) a(n)=sum(k=0, n, binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k));
vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */

Crossrefs

Cf. A005809, A001764, A006256, A006013, A045721, A188912, A188913.

Sequence in context: A175916 A231104 A085457 * A364923 A365192 A365187

Adjacent sequences: A188908 A188909 A188910 * A188912 A188913 A188914

Keyword

nonn,easy

Author

Emanuele Munarini, Apr 13 2011

Status

approved