A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.

0, 1, 1057, 68125, 1399325, 15227450, 110102426, 597639882, 2621915850, 9756511275, 31839011275, 93340522951, 250280856007, 622316813300, 1450471654100, 3196426654100, 6706824221076, 13476181309557, 26055415288725, 48670370285425, 88136930285425, 155187254126926

Offset

0,3

Comments

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

Links

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

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).

Formula

a(n) = n*(n+1)*(n+2)*(44*n^9 + 276*n^8 + 492*n^7 - 48*n^6 - 609*n^5 + 207*n^4 + 487*n^3 - 291*n^2 - 90*n + 60)/3168.

G.f.: x*(1 + 1044*x + 54462*x^2 + 595860*x^3 + 2048388*x^4 + 2563644*x^5 + 1193226*x^6 + 188508*x^7 + 7635*x^8 + 32*x^9)/(1-x)^13. - Robert Israel, Feb 18 2022

Maple

seq(n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168,
n=0..30); # Robert Israel, Feb 18 2022

Prog

(PARI) {a(n) = n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168};
(PARI) a(n) = sum(j=1, n, sum(i=1, j, i^5*j^5));

Crossrefs

Cf. A001296 (for power 1), A060493 (for squares), A346642 (for cubes), A351766 (for fourth powers).

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

Sequence in context: A223294 A035858 A120215 * A202611 A253678 A035760

Adjacent sequences: A351767 A351768 A351769 * A351771 A351772 A351773

Keyword

nonn,easy

Author

Roudy El Haddad, Feb 18 2022

Status

approved