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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..90
Programs
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Mathematica
Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k], {k,0,n}], {n,0,22}] -
Maxima
makelist(sum(binomial(n,k)*binomial(3*k,k)*binomial(3*n-3*k,n-k),k,0,n),n,0,12);
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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 */
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