OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..701
FORMULA
a(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(3*k,k).
Recurrence: (n+2)^2*a(n+2)-(28*n^2+85*n+64)*a(n+1)+3*(9*n^2+27*n+20)*a(n) = 0.
G.f.: F(1/3,2/3;1;27*x)/(1-x), where F(a1,a2;b1;z) is a hypergeometric series.
a(n) ~ 3^(3*n+7/2) / (52*Pi*n). - Vaclav Kotesovec, Mar 02 2014
a(n) = hypergeom([1/3, 2/3], [1], 27) - hypergeom([1, n+4/3, n+5/3], [n+2, n+2], 27)*multinomial(n+1, n+1, n+1). - Vladimir Reshetnikov, Oct 12 2016
MATHEMATICA
Table[Sum[Binomial[2k, k]Binomial[3k, k], {k, 0, n}], {n, 0, 16}]
Round@Table[Hypergeometric2F1[1/3, 2/3, 1, 27] - HypergeometricPFQ[{1, n + 4/3, n + 5/3}, {n + 2, n + 2}, 27] Multinomial[n + 1, n + 1, n + 1], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 12 2016 *)
Accumulate[Table[Binomial[2n, n]Binomial[3n, n], {n, 0, 20}]] (* Harvey P. Dale, Oct 27 2020 *)
PROG
(Maxima) makelist(sum(binomial(2*k, k)*binomial(3*k, k), k, 0, n), n, 0, 16);
(PARI) a(n) = sum(k=0, n, binomial(2*k, k)*binomial(3*k, k)); \\ Michel Marcus, Oct 13 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emanuele Munarini, Apr 14 2011
STATUS
approved