A188912 Binomial convolution of the binomial coefficients bin(3n,n)/(2n+1) (A001764).

1, 2, 8, 42, 260, 1816, 13962, 116094, 1029124, 9609144, 93569808, 942642696, 9763181946, 103455616400, 1117379189926, 12264816349938, 136501928050116, 1537591374945704, 17503603786398576, 201128739609458904, 2330480521265639136

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..93

Formula

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

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

From Vaclav Kotesovec, Jun 10 2019: (Start)

Recurrence: 8*n^2*(n+1)*(2*n+1)^2*(9*n^3-54*n^2+84*n-35)*a(n) = 24*n*(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*(n-1)*(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)^2*(n-1)*(3*n-7)*(3*n-5)*(9*n^3-27*n^2+3*n+4)*a(n-3).

a(n) ~ 3^(3*n + 1) / (Pi * n^3 * 2^(n + 1)). (End)

Mathematica

Table[Sum[Binomial[n, k]Binomial[3k, k]/(2k+1)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)/(2*k+1)*binomial(3*n-3*k, n-k)/(2*n-2*k+1), k, 0, n), n, 0, 12);

Crossrefs

Cf. A005809, A001764, A005809, A006256, A006013, A045721, A188911, A188913

Sequence in context: A013999 A130649 A054993 * A229285 A339460 A005315

Adjacent sequences: A188909 A188910 A188911 * A188913 A188914 A188915

Keyword

nonn,easy

Author

Emanuele Munarini, Apr 13 2011

Status

approved