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A354021 a(n) = Sum_{1 <= i < j < k < m <= n} (m*k*j*i)^2.

%I A354021 #16 Sep 19 2023 17:11:59
%S A354021 0,0,0,0,576,21076,296296,2475473,14739153,68943381,268880381,
%T A354021 909450751,2742417535,7522650135,19058554515,45123156390,100771975590,
%U A354021 213877057086,434042943246,846542846578,1593528150578
%N A354021 a(n) = Sum_{1 <= i < j < k < m <= n} (m*k*j*i)^2.
%C A354021 a(n) is the sum of all products of four distinct squares of positive integers up to n, i.e., the sum of all products of four distinct elements from the set of squares {1^2, ..., n^2}.
%H A354021 Winston de Greef, <a href="/A354021/b354021.txt">Table of n, a(n) for n = 0..10000</a>
%H A354021 Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.
%H A354021 Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.200-233">A generalization of multiple zeta value. Part 2: Multiple sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2) 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
%H A354021 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A354021 a(n) = Sum_{m=4..n} Sum_{k=3..m-1} Sum_{j=2..k-1} Sum_{i=1..j-1} (m*k*j*i)^2.
%F A354021 a(n) = n*(n+1)*(n-1)*(n-2)*(n-3)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(2*n - 5)*(5*n + 7)*(35*n^2 + 98*n + 72)/5443200.
%F A354021 a(n) = binomial(2*n+2,9)*(5*n + 7)*(35*n^2 + 98*n + 72)/(5!*4).
%o A354021 (PARI) {a(n) = n*(n + 1)*(n - 1)*(n - 2)*(n - 3)*(2*n + 1)*(2*n - 1)*(2*n - 3)*(2*n - 5)*(5*n + 7)*(35*n^2 + 98*n + 72)/5443200};
%Y A354021 Cf. A353021 (for nondistinct squares).
%Y A354021 Cf. A000290 (squares), A000330 (sum of squares), A000596 (for two squares),  A000597 (for three squares).
%Y A354021 Cf. A001298 (for power 1).
%K A354021 nonn,easy
%O A354021 0,5
%A A354021 _Roudy El Haddad_, May 14 2022