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
KEYWORD
nonn,easy
AUTHOR
Roudy El Haddad, May 14 2022
STATUS
approved