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A352979
a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (k*j*i)^3.
2
0, 1, 585, 28800, 505280, 4951530, 33209946, 170320080, 714724560, 2566030995, 8130545995, 23253835176, 61054704360, 149085989780, 342048076020, 743408003520, 1540821690816, 3062326169925, 5862986735085, 10855192630480, 19500255870480
OFFSET
0,3
COMMENTS
a(n) is the sum of all products of three cubes of positive integers up to n, i.e., the sum of all products of three elements from the set of cubes {1^3, ..., n^3}.
REFERENCES
El Haddad, R. (2022). A generalization of multiple zeta value. Part 1: Recurrent sums. Notes on Number Theory and Discrete Mathematics, 28(2), 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1). [Typo corrected by Georg Fischer, Sep 30 2022]
FORMULA
a(n) = n^2 * (n + 1)^2 * (n + 2) * (n + 3) * (35*n^6 + 205*n^5 + 263*n^4 - 221*n^3 - 214*n^2 + 324*n - 112)/13440.
a(n) = binomial(n+3,4)*binomial(n+1,2)*(35*n^6 + 205*n^5 + 263*n^4 - 221*n^3 - 214*n^2 + 324*n - 112)/280.
PROG
(PARI) {a(n) = n^2 * (n + 1)^2 * (n + 2) * (n + 3) * (35*n^6 + 205*n^5 + 263*n^4 - 221*n^3 - 214*n^2 + 324*n - 112)/13440};
(Python)
def A352979(n): return n**2*(n*(n*(n*(n*(n*(n*(n*(n*(n*(35*n + 450) + 2293) + 5700) + 6405) + 770) - 3661) - 240) + 2320) + 40) - 672)//13440 # Chai Wah Wu, May 14 2022
CROSSREFS
Cf. A352980 (for distinct cubes).
Cf. A001297 (for power 1), A351105 (for squares).
Cf. A000578 (cubes), A000537 (sum of first n cubes), A346642 (order 2).
Sequence in context: A038486 A024380 A258331 * A022243 A217446 A369861
KEYWORD
nonn,easy
AUTHOR
Roudy El Haddad, Apr 13 2022
STATUS
approved