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