This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188682 #17 May 22 2025 18:37:08
%S A188682 1,4,49,1057,28282,848101,27357493,928760053,32747441926,
%T A188682 1188869998801,44174723634526,1672716549215326,64340599136306926,
%U A188682 2507814491482180894,98859670298036582494,3935425516392739090270,158006444406545953115743
%N A188682 Partial sums of binomials bin(3n,n)^2/(2n+1).
%H A188682 Harvey P. Dale, <a href="/A188682/b188682.txt">Table of n, a(n) for n = 0..606</a>
%F A188682 a(n) = sum(bin(3*k,k)^2/(2*k+1),k=0..n).
%F A188682 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.
%F A188682 a(n) ~ 3^(6*n+7)/(713*Pi*n^2*2^(4*n+3)). - _Vaclav Kotesovec_, Aug 06 2013
%t A188682 Table[Sum[Binomial[3k,k]^2/(2k+1),{k,0,n}],{n,0,20}]
%t A188682 Accumulate[Table[Binomial[3n,n]^2/(2n+1),{n,0,20}]] (* _Harvey P. Dale_, Jul 10 2016 *)
%o A188682 (Maxima) makelist(sum(binomial(3*k,k)^2/(2*k+1),k,0,n),n,0,20);
%Y A188682 Cf. A005809, A001764, A188676, A104859, A188678, A188679, A188680, A188681, A188683, A188684, A188685, A188686, A188687.
%K A188682 nonn,easy
%O A188682 0,2
%A A188682 _Emanuele Munarini_, Apr 08 2011