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
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..105
Programs
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Mathematica
Table[Sum[Binomial[n,k]Binomial[3k,k]Binomial[3n-3k,n-k]/(2n-2k+1), {k,0,n}], {n,0,22}] -
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);
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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)/(2*n-2*k+1)); vector(66, n, a(n-1)) /* show terms */ /* Joerg Arndt, Apr 13 2011 */
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