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%I #17 Sep 26 2019 11:32:33
%S 1,10,235,7291,252316,9270325,353892421,13874930821,554792522662,
%T 22521121103287,925224047453512,38381686035811912,1605078750713101912,
%U 67578873844051699048,2861782692234129345448,121795323921169907086504
%N Partial sums of binomial(3n,n)^2.
%H Seiichi Manyama, <a href="/A188679/b188679.txt">Table of n, a(n) for n = 0..604</a>
%F a(n) = sum(C(3k,k)^2, k=0..n).
%F Recurrence: 4*(2*n^2+7*n+6)^2 * a(n+2) -(745*n^4+4486*n^3+10093*n^2 +10056*n+3744) * a(n+1) +9*(9*n^2+27*n+20)^2 * a(n) = 0.
%F G.f.: (1-x)^(-1)*F(1/3,1/3,2/3,2/3;1/2,1/2,1;729*x/16), where F(a1,a2,a3,a4;b1,b2,b3;z) is a hypergeometric series.
%F a(n) ~ 3^(6*n+7)/(713*Pi*n*2^(4*n+2)). - _Vaclav Kotesovec_, Aug 06 2013
%t Table[Sum[Binomial[3k,k]^2,{k,0,n}],{n,0,20}]
%t Accumulate[Table[Binomial[3n,n]^2,{n,0,20}]] (* _Harvey P. Dale_, Sep 26 2019 *)
%o (Maxima) makelist(sum(binomial(3*k,k)^2,k,0,n),n,0,20);
%Y Cf. A005809, A001764, A188676, A104859, A188678, A188680, A188681, A188682, A188683, A188684, A188685, A188686, A188687.
%K nonn,easy
%O 0,2
%A _Emanuele Munarini_, Apr 08 2011
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