OFFSET
0,2
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*binomial(3*k,k).
Recurrence: (n+3)^2*a(n+3)-(30*n^2+150*n+187)*a(n+2)+57*(n+2)^2*a(n+1)-28*(n+1)*(n+2)*a(n)=0.
E.g.f.: exp(x)*F(1/3,2/3;1,1;27*x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) = hypergeom([1/3, 2/3, -n], [1, 1], -27). - Vladimir Reshetnikov, Oct 15 2017
a(n) ~ 2^(2*n+1) * 7^(n+1) / (3^(5/2)*Pi*n). - Vaclav Kotesovec, Nov 27 2017
MATHEMATICA
Table[Sum[Binomial[n, k]Binomial[2k, k]Binomial[3k, k], {k, 0, n}], {n, 0, 16}]
Table[HypergeometricPFQ[{1/3, 2/3, -n}, {1, 1}, -27], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 15 2017 *)
PROG
(Maxima) makelist(sum(binomial(n, k)*binomial(2*k, k)*binomial(3*k, k), k, 0, n), n, 0, 16);
(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(2*k, k)*binomial(3*k, k)); \\ Michel Marcus, Oct 15 2017
(Magma) [&+[Binomial(n, k)*Binomial(2*k, k)*Binomial(3*k, k): k in [0..n]]: n in [0.. 18]]; // Vincenzo Librandi, Oct 16 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, Apr 14 2011
STATUS
approved