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

0,4

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

Table of n, a(n) for n=0..21.

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]

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.

Prog

(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

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).

Sequence in context: A269197 A259831 A269105 * A223359 A046234 A229408

Adjacent sequences: A352977 A352978 A352979 * A352981 A352982 A352983

Keyword

nonn,easy

Author

Roudy El Haddad, Apr 13 2022

Status

approved