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
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, Apr 13 2011
STATUS
approved