OFFSET
0,3
COMMENTS
a(n) is the sum of all products of two cubes of positive integers up to n, i.e., the sum of all products of two elements from the set of cubes {1^3, ..., n^3}.
LINKS
Roudy El Haddad, Recurrent Sums and Partition Identities, arXiv:2101.09089 [math.NT], 2021.
Roudy El Haddad, A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).
FORMULA
a(n) = n*(n+1)*(n+2)*(21*n^5+69*n^4+45*n^3-21*n^2-6*n+4)/672 (from the recurrent form of Faulhaber's formula).
G.f.: -(8*x^5+179*x^4+584*x^3+424*x^2+64*x+1)*x/(x-1)^9. - Alois P. Heinz, Jan 27 2022
EXAMPLE
For n=3,
a(3) = (1*1)^3+(2*1)^3+(2*2)^3+(3*1)^3+(3*2)^3+(3*3)^3 = 1045,
a(3) = 1^3*(1^3)+2^3*(1^3+2^3)+3^3*(1^3+2^3+3^3) = 1045.
MATHEMATICA
CoefficientList[Series[-(8 x^5 + 179 x^4 + 584 x^3 + 424 x^2 + 64 x + 1) x/(x - 1)^9, {x, 0, 26}], x] (* Michael De Vlieger, Feb 04 2022 *)
PROG
(PARI) {a(n) = n*(n+1)*(n+2)*(21n^5+69n^4+45n^3-21n^2-6n+4)/672};
(PARI) a(n) = sum(i=1, n, sum(j=1, i, i^3*j^3)); \\ Michel Marcus, Jan 27 2022
(Python)
def A346642(n): return n*(n**2*(n*(n*(n*(n*(21*n + 132) + 294) + 252) + 21) - 56) + 8)//672 # Chai Wah Wu, Feb 17 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Roudy El Haddad, Jan 24 2022
STATUS
approved