A351805 a(n) = Sum_{1 <= i < j <= n} j^5*i^5.

0, 0, 32, 8051, 290675, 4353175, 38761975, 243824182, 1194358326, 4842169350, 16924669350, 52488756425, 147511725257, 381689190701, 920589376525, 2089893985900, 4500779925100, 9254143113132, 18262909865676, 34746798604575, 63973358604575, 114343801467875

Offset: 0

Author

Roudy El Haddad, Feb 19 2022

Keywords

• nonn
• easy

Comments

a(n) is the sum of all products of two distinct elements from the set {1^5, ..., n^5}.

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 (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Crossrefs

Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares), A347107 (for cubes), (for fourth powers).

Cf. A000584 (fifth powers), A000539 (sum of fifth powers).

Programs

PARI

{a(n) = n*(n-1)*(n+1)*(44*n^9+120*n^8-132*n^7-540*n^6+99*n^5+912*n^4-11*n^3-672*n^2+120)/3168};

Formula

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^5*i^5.

a(n) = n*(n - 1)*(n + 1)*(44*n^9 + 120*n^8 - 132*n^7 - 540*n^6 + 99*n^5 + 912*n^4 - 11*n^3 - 672*n^2 + 120)/3168.

G.f.: -x^2*(x^9 +1044*x^8 +54462*x^7 +595860*x^6 +2048388*x^5 +2563644*x^4 +1193226*x^3 +188508*x^2 +7635*x +32)/(x-1)^13. - Alois P. Heinz, Feb 19 2022