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A188685 Partial alternating sums of binomial(3n,n)^2/(2n+1)^2.

%I A188685 #22 May 22 2025 21:01:26
%S A188685 1,0,9,135,2890,71639,1967545,58125959,1813561210,59034994415,
%T A188685 1987910416810,68818255912790,2437897047570874,88061136002276310,
%U A188685 3234416650430634090,120525771933269446806,4548292982313797644875
%N A188685 Partial alternating sums of binomial(3n,n)^2/(2n+1)^2.
%H A188685 Seiichi Manyama, <a href="/A188685/b188685.txt">Table of n, a(n) for n = 0..608</a>
%F A188685 a(n) = Sum_{k=0..n} (-1)^(n-k)*A001764(k)^2.
%F A188685 4*(2*n^2 + 9*n + 10)^2*a(n+2) - (713*n^4 + 4230*n^3 + 9317*n^2 + 9000*n + 3200)*a(n+1) - 9*(9*n^2 + 27*n + 20)^2*a(n) = 0.
%F A188685 a(n) ~ 3^(6*n+7)/(745*Pi*n^3*2^(4*n+4)). - _Vaclav Kotesovec_, Aug 06 2013
%p A188685 A001764 := proc(n) binomial(3*n,n)/(2*n+1) ; end proc:
%p A188685 A188685 := proc(n) add( (-1)^(n-k)*A001764(k)^2,k=0..n) ; end proc: # _R. J. Mathar_, Apr 11 2011
%t A188685 Table[Sum[Binomial[3k,k]^2(-1)^(n-k)/(2k+1)^2,{k,0,n}],{n,0,20}]
%o A188685 (Maxima) makelist(sum(binomial(3*k,k)^2*(-1)^(n-k)/(2*k+1)^2,k,0,n),n,0,20);
%o A188685 (Magma) [ &+[(-1)^(n-k)*Binomial(3*k, k)^2/(2*k+1)^2: k in [0..n]]: n in [0..16]];  // _Bruno Berselli_, Apr 11 2011
%Y A188685 Cf. A001764, A005809, A104859, A188678, A188681.
%Y A188685 Cf. Alternate partial sums of binomial(3n,n)^2/(2n+1)^k: A188680 (k=0), A188683 (k=1), this sequence (k=2).
%K A188685 nonn,easy
%O A188685 0,3
%A A188685 _Emanuele Munarini_, Apr 08 2011