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A188684 Partial sums of binomials binomial(3n,n)^2/(2n+1)^2.

%I A188684 #31 May 22 2025 18:37:48
%S A188684 1,2,11,155,3180,77709,2116893,62210397,1933897566,62782453191,
%T A188684 2109727864416,72915894194016,2579631197677680,93078664247524864,
%U A188684 3415556450680435264,127175745034380516160,4795994499281447607841
%N A188684 Partial sums of binomials binomial(3n,n)^2/(2n+1)^2.
%H A188684 Seiichi Manyama, <a href="/A188684/b188684.txt">Table of n, a(n) for n = 0..608</a>
%F A188684 a(n) = sum( A001764(k)^2 , k=0..n).
%F A188684 4*(2*n^2+9*n+10)^2*a(n+2) - (745*n^4+4518*n^3+10285*n^2+10440*n+4000)*a(n+1) + 9*(9*n^2+27*n+20)^2*a(n) = 0.
%F A188684 a(n) = 4F3(1/3,1/3,2/3,2/3; 1,3/2,3/2; 729/16) - Gamma^2(3n+4) *5F4(1,n+4/3,n+4/3,n+5/3,n+5/3; n+2,n+2,n+5/2,n+5/2; 729/16)/ (Gamma(n+2)*Gamma(2n+3))^2, with pFq() generalized hypergeometric functions. - _Charles R Greathouse IV_, Apr 14 2011
%F A188684 a(n) ~ 3^(6*n+7)/(713*Pi*n^3*2^(4*n+4)). - _Vaclav Kotesovec_, Aug 06 2013
%t A188684 Table[Sum[Binomial[3k,k]^2/(2k+1)^2,{k,0,n}],{n,0,20}]
%o A188684 (Maxima) makelist(sum(binomial(3*k,k)^2/(2*k+1)^2,k,0,n),n,0,20);
%o A188684 (Magma) [&+[Binomial(3*k,k)^2/(2*k+1)^2: k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Nov 04 2016
%Y A188684 Cf. A005809, A001764, A188681.
%Y A188684 Cf. Partial sums of binomial(3n,n)^2/(2n+1)^k: A188679 (k=0), A188682 (k=1), this sequence (k=2).
%Y A188684 Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), A188683 (k=1), A188685 (k=2).
%K A188684 nonn,easy
%O A188684 0,2
%A A188684 _Emanuele Munarini_, Apr 08 2011