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

0,3

Comments

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

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.

Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

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

Prog

(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};

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).

Sequence in context: A069052 A334604 A342455 * A221614 A086752 A248001

Adjacent sequences: A351802 A351803 A351804 * A351806 A351807 A351808

Keyword

nonn,easy

Author

Roudy El Haddad, Feb 19 2022

Status

approved