A347107 a(n) = Sum_{1 <= i < j <= n} j^3*i^3.

0, 0, 8, 251, 2555, 15055, 63655, 214918, 616326, 1561110, 3586110, 7612385, 15139553, 28506101, 51229165, 88438540, 147420940, 238291788, 374813076, 575377095, 864177095, 1272587195, 1840775123, 2619572626, 3672629650, 5078879650, 6935344650, 9360309933

Offset

0,3

Comments

a(n) is the sum of all products of two distinct cubes of positive integers up to n, i.e., the sum of all products of two distinct elements from the set of cubes {1^3, ..., n^3}.

Links

Table of n, a(n) for n=0..27.

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 (9,-36,84,-126,126,-84,36,-9,1).

Formula

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^3*i^3.

a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672 (from the generalized form of Faulhaber's formula).

From Alois P. Heinz, Jan 27 2022: (Start)

a(n) = Sum_{i=1..n} A000578(i)*A000537(i-1) = Sum_{i=1..n} i^3*(i*(i-1)/2)^2.

G.f.: -(x^5+64*x^4+424*x^3+584*x^2+179*x+8)*x^2/(x-1)^9. (End)

Example

For n=3, a(3) = (2*1)^3+(3*1)^3+(3*2)^3 = 251.

Mathematica

CoefficientList[Series[-(x^5 + 64 x^4 + 424 x^3 + 584 x^2 + 179 x + 8) x^2/(x - 1)^9, {x, 0, 27}], x] (* Michael De Vlieger, Feb 04 2022 *)

Prog

(PARI) a(n) = sum(i=2, n, sum(j=1, i-1, i^3*j^3));
(PARI) {a(n) = n*(n+1)*(n-1)*(21n^5+36n^4-21n^3-48n^2+8)/672};
(Python)
def A347107(n): return n*(n**2*(n*(n*(n*(n*(21*n + 36) - 42) - 84) + 21) + 56) - 8)//672 # Chai Wah Wu, Feb 17 2022

Crossrefs

Cf. A000537, A000578.

Cf. A346642 (for nondistinct cubes).

Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares).

Sequence in context: A221518 A317519 A300202 * A138323 A214817 A303415

Adjacent sequences: A347104 A347105 A347106 * A347108 A347109 A347110

Keyword

nonn,easy

Author

Roudy El Haddad, Jan 27 2022

Status

approved