A352980 a(n) = Sum_{1 <= i < j < k <= n} (k*j*i)^3.

0, 0, 0, 216, 16280, 335655, 3587535, 25421200, 135459216, 584760870, 2145870870, 6918983280, 20073184560, 53334782501, 131555523645, 304453955520, 666698215360, 1390977293580, 2780695001196, 5351537889480, 9954554649480, 17957698726275

Offset: 0

Author

Roudy El Haddad, Apr 13 2022

Keywords

• nonn
• easy

Comments

a(n) is the sum of all products of three distinct cubes of positive integers up to n, i.e., the sum of all products of three 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. See Theorem 5.1 for m = 3 and p = 3.
• Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). [Typo corrected by _Georg Fischer_, Sep 30 2022]

Crossrefs

Cf. A352979 (for nondistinct cubes).

Cf. A001303 (for power 1), A000597 (for squares).

Cf. A000578 (cubes), A000537 (sum of first n cubes), A347107 (order 2).

Programs

PARI

{a(n) = n^2 * (n + 1)^2 * (n - 1) * (n - 2) * (35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/13440};

Python

def A352980(n): return n**2*(n*(n*(n*(n*(n*(n*(n*(n*(n*(35*n - 30) - 347) + 180) + 1365) - 350) - 2541) + 240) + 2160) - 40) - 672)//13440 # Chai Wah Wu, May 15 2022

Formula

a(n) = Sum_{k=3..n} Sum_{j=2..k-1} Sum_{i=1..j-1} k^3*j^3*i^3.

a(n) = n^2 * (n + 1)^2 * (n - 1) * (n - 2) * (35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/13440.

a(n) = binomial(n+1,4)*binomial(n+1,2)*(35*n^6 + 5*n^5 - 237*n^4 - 77*n^3 + 502*n^2 + 148*n -336)/280.