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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..93
Programs
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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}] -
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);
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)