A188913 Binomial convolution of the binomial coefficients bin(3n,n) (A005809) and bin(3n,n)/(2n+1) (A001764).

1, 4, 24, 168, 1300, 10896, 97734, 928752, 9262116, 96091440, 1029267888, 11311712352, 126921365298, 1448378629600, 16760687848890, 196237061599008, 2320532776851972, 27676644749022672, 332568471941572944, 4022574792189178080

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..105

Formula

a(n) = sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k)/(2*n-2*k+1),k=0..n).

E.g.f.: F(1/3,2/3;1/2,1;27*x/4)*F(1/3,2/3;1,3/2;27*x/4), 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 + 84*n - 35)*a(n) = 24*(324*n^7 - 2187*n^6 + 4689*n^5 - 4185*n^4 + 1464*n^3 + 122*n^2 - 223*n + 44)*a(n-1) - 18*(3645*n^7 - 30618*n^6 + 96066*n^5 - 144585*n^4 + 103662*n^3 - 21834*n^2 - 10860*n + 4480)*a(n-2) + 2187*(n-2)*(n-1)*(3*n-7)*(3*n-5)*(9*n^3 - 27*n^2 + 3*n + 4)*a(n-3). - Vaclav Kotesovec, Feb 25 2014

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

Mathematica

Table[Sum[Binomial[n, k]Binomial[3k, k]Binomial[3n-3k, n-k]/(2n-2k+1), {k, 0, n}], {n, 0, 22}]

Prog

(Maxima) makelist(sum(binomial(n, k)*binomial(3*k, k)*binomial(3*n-3*k, n-k)/(2*n-2*k+1), 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)/(2*n-2*k+1));
vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */

Crossrefs

Cf. A005809, A001764, A006256, A006013, A045721, A188911, A188912.

Sequence in context: A369503 A212277 A246423 * A364324 A052685 A221088

Adjacent sequences: A188910 A188911 A188912 * A188914 A188915 A188916

Keyword

nonn,easy

Author

Emanuele Munarini, Apr 13 2011

Status

approved