A188441 Partial sums of binomial(2n,n)*binomial(3n,n) (A006480).

1, 7, 97, 1777, 36427, 793183, 17946319, 417019279, 9882531049, 237755962549, 5788752753889, 142315748216929, 3527047510738129, 88005145583604529, 2208577811494332529, 55703557596868964209, 1411049022002884046539

Offset: 0

Author

Emanuele Munarini, Apr 14 2011

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..701

Crossrefs

Cf. A006480.

Programs

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 *)

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

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