This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347107 #73 Jul 07 2025 15:30:25
%S A347107 0,0,8,251,2555,15055,63655,214918,616326,1561110,3586110,7612385,
%T A347107 15139553,28506101,51229165,88438540,147420940,238291788,374813076,
%U A347107 575377095,864177095,1272587195,1840775123,2619572626,3672629650,5078879650,6935344650,9360309933
%N A347107 a(n) = Sum_{1 <= i < j <= n} j^3*i^3.
%C A347107 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}.
%H A347107 Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.
%H A347107 Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.200-233">A generalization of multiple zeta value. Part 2: Multiple sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
%H A347107 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A347107 a(n) = Sum_{j=2..n} Sum_{i=1..j-1} j^3*i^3.
%F A347107 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).
%F A347107 From _Alois P. Heinz_, Jan 27 2022: (Start)
%F A347107 a(n) = Sum_{i=1..n} A000578(i)*A000537(i-1) = Sum_{i=1..n} i^3*(i*(i-1)/2)^2.
%F A347107 G.f.: -(x^5+64*x^4+424*x^3+584*x^2+179*x+8)*x^2/(x-1)^9. (End)
%e A347107 For n=3, a(3) = (2*1)^3+(3*1)^3+(3*2)^3 = 251.
%t A347107 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 *)
%t A347107 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,8,251,2555,15055,63655,214918,616326},30] (* _Harvey P. Dale_, Jul 07 2025 *)
%o A347107 (PARI) a(n) = sum(i=2, n, sum(j=1, i-1, i^3*j^3));
%o A347107 (PARI) {a(n) = n*(n+1)*(n-1)*(21*n^5+36*n^4-21*n^3-48*n^2+8)/672};
%o A347107 (Python)
%o A347107 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
%Y A347107 Cf. A000537, A000578.
%Y A347107 Cf. A346642 (for nondistinct cubes).
%Y A347107 Cf. A000217 (for power 0), A000914 (for power 1), A000596 (for squares).
%K A347107 nonn,easy
%O A347107 0,3
%A A347107 _Roudy El Haddad_, Jan 27 2022