cp's OEIS Frontend

A346642 a(n) = Sum_{j=1..n} Sum_{i=1..j} j^3*i^3.

%I A346642 #70 May 28 2022 12:42:40
%S A346642 0,1,73,1045,7445,35570,130826,399738,1063290,2539515,5564515,
%T A346642 11362351,21875503,40068860,70321460,118921460,194681076,309689493,
%U A346642 480223005,727832905,1080632905,1574809126,2256376958,3183210350,4427370350,6077760975,8243141751
%N A346642 a(n) = Sum_{j=1..n} Sum_{i=1..j} j^3*i^3.
%C A346642 a(n) is the sum of all products of two cubes of positive integers up to n, i.e., the sum of all products of two elements from the set of cubes {1^3, ..., n^3}.
%H A346642 Roudy El Haddad, <a href="https://arxiv.org/abs/2101.09089">Recurrent Sums and Partition Identities</a>, arXiv:2101.09089 [math.NT], 2021.
%H A346642 Roudy El Haddad, <a href="https://doi.org/10.7546/nntdm.2022.28.2.167-199">A generalization of multiple zeta value. Part 1: Recurrent sums</a>. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 167-199, DOI: 10.7546/nntdm.2022.28.2.167-199.
%H A346642 <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 A346642 a(n) = n*(n+1)*(n+2)*(21*n^5+69*n^4+45*n^3-21*n^2-6*n+4)/672 (from the recurrent form of Faulhaber's formula).
%F A346642 G.f.: -(8*x^5+179*x^4+584*x^3+424*x^2+64*x+1)*x/(x-1)^9. - _Alois P. Heinz_, Jan 27 2022
%e A346642 For n=3,
%e A346642 a(3) = (1*1)^3+(2*1)^3+(2*2)^3+(3*1)^3+(3*2)^3+(3*3)^3 = 1045,
%e A346642 a(3) = 1^3*(1^3)+2^3*(1^3+2^3)+3^3*(1^3+2^3+3^3) = 1045.
%t A346642 CoefficientList[Series[-(8 x^5 + 179 x^4 + 584 x^3 + 424 x^2 + 64 x + 1) x/(x - 1)^9, {x, 0, 26}], x] (* _Michael De Vlieger_, Feb 04 2022 *)
%o A346642 (PARI) {a(n) = n*(n+1)*(n+2)*(21n^5+69n^4+45n^3-21n^2-6n+4)/672};
%o A346642 (PARI) a(n) = sum(i=1, n, sum(j=1, i, i^3*j^3)); \\ _Michel Marcus_, Jan 27 2022
%o A346642 (Python)
%o A346642 def A346642(n): return n*(n**2*(n*(n*(n*(n*(21*n + 132) + 294) + 252) + 21) - 56) + 8)//672 # _Chai Wah Wu_, Feb 17 2022
%Y A346642 Cf. A000537 (sum of first n cubes), A347107 (for distinct cubes).
%Y A346642 Cf. A001296 (for power 1), A060493 (for squares).
%K A346642 nonn,easy
%O A346642 0,3
%A A346642 _Roudy El Haddad_, Jan 24 2022