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

0, 0, 0, 0, 576, 21076, 296296, 2475473, 14739153, 68943381, 268880381, 909450751, 2742417535, 7522650135, 19058554515, 45123156390, 100771975590, 213877057086, 434042943246, 846542846578, 1593528150578

Offset

0,5

Comments

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}.

Links

Winston de Greef, Table of n, a(n) for n = 0..10000

Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.

Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2) 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Formula

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.

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.

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

Prog

(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};

Crossrefs

Cf. A353021 (for nondistinct squares).

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

Cf. A001298 (for power 1).

Sequence in context: A330840 A226285 A370226 * A282780 A268638 A230522

Adjacent sequences: A354018 A354019 A354020 * A354022 A354023 A354024

Keyword

nonn,easy

Author

Roudy El Haddad, May 14 2022

Status

approved