%I #16 Sep 19 2023 17:11:59

%S 0,0,0,0,576,21076,296296,2475473,14739153,68943381,268880381,

%T 909450751,2742417535,7522650135,19058554515,45123156390,100771975590,

%U 213877057086,434042943246,846542846578,1593528150578

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

%C 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 Winston de Greef, <a href="/A354021/b354021.txt">Table of n, a(n) for n = 0..10000</a>

%H Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.

%H 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 <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 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 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 a(n) = binomial(2*n+2,9)*(5*n + 7)*(35*n^2 + 98*n + 72)/(5!*4).

%o (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 Cf. A353021 (for nondistinct squares).

%Y Cf. A000290 (squares), A000330 (sum of squares), A000596 (for two squares), A000597 (for three squares).

%Y Cf. A001298 (for power 1).

%K nonn,easy

%O 0,5

%A _Roudy El Haddad_, May 14 2022