OFFSET
0,3
COMMENTS
a(n) is the sum of all products of two distinct cubes of positive integers up to n, i.e., the sum of all products of two distinct elements from the set of cubes {1^3, ..., n^3}.
LINKS
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 (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^3*i^3.
a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672 (from the generalized form of Faulhaber's formula).
From Alois P. Heinz, Jan 27 2022: (Start)
G.f.: -(x^5+64*x^4+424*x^3+584*x^2+179*x+8)*x^2/(x-1)^9. (End)
EXAMPLE
For n=3, a(3) = (2*1)^3+(3*1)^3+(3*2)^3 = 251.
MATHEMATICA
CoefficientList[Series[-(x^5 + 64 x^4 + 424 x^3 + 584 x^2 + 179 x + 8) x^2/(x - 1)^9, {x, 0, 27}], x] (* Michael De Vlieger, Feb 04 2022 *)
PROG
(PARI) a(n) = sum(i=2, n, sum(j=1, i-1, i^3*j^3));
(PARI) {a(n) = n*(n+1)*(n-1)*(21n^5+36n^4-21n^3-48n^2+8)/672};
(Python)
def A347107(n): return n*(n**2*(n*(n*(n*(n*(21*n + 36) - 42) - 84) + 21) + 56) - 8)//672 # Chai Wah Wu, Feb 17 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roudy El Haddad, Jan 27 2022
STATUS
approved