A353021 - OEIS

%I #32 May 15 2026 13:51:12

%S 0,1,341,13013,196053,1733303,10787231,52253971,209609235,725520510,

%T 2230238010,6217887390,15973440990,38276304066,86383520146,

%U 185042663146,378620563178,743881306623,1409531082531,2585397711611,4605062303611

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

%C a(n) is the sum of all products of four squares of positive integers up to n, i.e., the sum of all products of four elements from the set of squares {1^2, ..., n^2}.

%H Roudy El Haddad, <a href="https://arxiv.org/abs/2101.09089">Recurrent Sums and Partition Identities</a>, arXiv:2101.09089 [math.NT], 2021.

%H Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.167-199">A generalization of multiple zeta value. Part 1: Recurrent sums</a>, Notes Number Theory and Disc. Math. 28(2) (2022), 167-199. See Theorem 4.8 for m = 4 and p = 2.

%H Andrii Husiev, <a href="https://arxiv.org/abs/2605.06689">Extended Central Factorial Numbers and the Flickering Operator</a>, arXiv:2605.06689 [math.GM], 2026. See p. 5.

%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) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(2*n + 1)*(2*n + 3)*(2*n + 5)*(2*n + 7)*(5*n - 2)*(35*n^2 - 28*n + 9)/5443200.

%F a(n) = binomial(2*n+8,9)*(5*n - 2)*(35*n^2 - 28*n + 9)/(5!*4).

%o (PARI) {a(n) = n*(n + 1)*(n + 2)*(n + 3)*(n + 4)*(2*n + 1)*(2*n + 3)*(2*n + 5)*(2*n + 7)*(5*n - 2)*(35*n^2 - 28*n + 9)/5443200};

%o (Python)

%o def A353021(n): return n*(n*(n*(n*(n*(n*(n*(n*(8*n*(n*(70*n*(5*n + 84) + 40417) + 144720) + 2238855) + 2050020) + 207158) - 810600) - 58505) + 322740) + 7956) - 45360)//5443200 # _Chai Wah Wu_, May 14 2022

%Y Cf. A354021 (for distinct squares).

%Y Cf. A000290 (squares), A000330 (sum of squares), A060493 (for two squares), A351105 (for three squares).

%Y Cf. A000915 (for power 1).

%K nonn,easy

%O 0,3

%A _Roudy El Haddad_, Apr 17 2022