A188682 Partial sums of binomials bin(3n,n)^2/(2n+1).

1, 4, 49, 1057, 28282, 848101, 27357493, 928760053, 32747441926, 1188869998801, 44174723634526, 1672716549215326, 64340599136306926, 2507814491482180894, 98859670298036582494, 3935425516392739090270, 158006444406545953115743

Offset: 0

Author

Emanuele Munarini, Apr 08 2011

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..606

Crossrefs

Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188681, A188683, A188684, A188685, A188686, A188687.

Programs

Mathematica

Table[Sum[Binomial[3k,k]^2/(2k+1),{k,0,n}],{n,0,20}]
Accumulate[Table[Binomial[3n,n]^2/(2n+1),{n,0,20}]] (* Harvey P. Dale, Jul 10 2016 *)

Maxima

makelist(sum(binomial(3*k,k)^2/(2*k+1),k,0,n),n,0,20);

Formula

a(n) = sum(bin(3*k,k)^2/(2*k+1),k=0..n).

Recurrence: 4*(n+2)^2*(4*n^2+16*n+15) * a(n+2) -(745*n^4+4502*n^3+10181*n^2+10216*n+3840) * a(n+1) +9*(9*n^2+27*n+20)^2 *a(n) = 0.

a(n) ~ 3^(6*n+7)/(713*Pi*n^2*2^(4*n+3)). - Vaclav Kotesovec, Aug 06 2013