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

Author

Roudy El Haddad, Feb 18 2022

Keywords

• nonn
• easy

Comments

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

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

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).

Programs

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)[phone])+1)+30)//1800 # Chai Wah Wu, Oct 03 2024

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