A351760 a(n) = Sum_{1 <= i < j <= n} (i*j)^4.

0, 0, 16, 1393, 26481, 247731, 1516515, 6978790, 26131686, 83684778, 237014778, 607915231, 1436816095, 3170754405, 6600189141, 13064343516, 24750198748, 45116627556, 79482515700, 135826148445, 225852708445, 366397514791, 581244702423, 903454469346, 1378306878690, 2066986566190

Offset

0,3

Comments

a(n) is the sum of all products of two distinct elements from the set {1^4, ..., n^4}.

Links

Table of n, a(n) for n=0..25.

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 (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Formula

a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^4*i^4.

a(n) = n*(n - 1)*(n + 1)*(2*n - 1)*(2*n + 1)*(9*n^5 + 20*n^4 - 15*n^3 - 50*n^2 + n + 30)/1800.

a(n) = binomial(2*n+2, 5)*(9*n^5 + 20*n^4 - 15*n^3 - 50*n^2 + n + 30)/5!.

G.f.: x^2*(16 + 1217*x + 12038*x^2 + 30415*x^3 + 23364*x^4 + 5263*x^5 + 262*x^6 + x^7)/(1 - x)^11. - Stefano Spezia, Feb 18 2022

Prog

(PARI) {a(n) = n*(n-1)*(n+1)*(2*n-1)*(2*n+1)*(9*n^5+20*n^4-15*n^3-50*n^2+n+30)/1800};
(PARI) a(n) = sum(j=2, n, sum(i=1, j-1, i^4*j^4));
(Python)
def A351760(n): return n*(n*(n*(n*(n*(n*(n*(n*(n*(9*n+20<<2)-105)-300)+88)+390)-20)-200)+1)+30)//1800 # Chai Wah Wu, Oct 03 2024

Crossrefs

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

Cf. A000583 (fourth powers), A000538 (sum of fourth powers).

Sequence in context: A321583 A057994 A221607 * A330335 A160251 A106176

Adjacent sequences: A351757 A351758 A351759 * A351761 A351762 A351763

Keyword

nonn,easy

Author

Roudy El Haddad, Feb 18 2022

Status

approved