cp's OEIS Frontend

A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.

%I A351770 #24 Jun 03 2022 11:17:40
%S A351770 0,1,1057,68125,1399325,15227450,110102426,597639882,2621915850,
%T A351770 9756511275,31839011275,93340522951,250280856007,622316813300,
%U A351770 1450471654100,3196426654100,6706824221076,13476181309557,26055415288725,48670370285425,88136930285425,155187254126926
%N A351770 a(n) = Sum_{j=1..n} Sum_{i=1..j} (i*j)^5.
%C A351770 a(n) is the sum of all products of two elements from the set {1^5, ..., n^5}.
%H A351770 Roudy El Haddad, <a href="https://arxiv.org/abs/2101.09089">Recurrent Sums and Partition Identities</a>, arXiv:2101.09089 [math.NT], 2021.
%H A351770 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 A351770 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A351770 a(n) = n*(n+1)*(n+2)*(44*n^9 + 276*n^8 + 492*n^7 - 48*n^6 - 609*n^5 + 207*n^4 + 487*n^3 - 291*n^2 - 90*n + 60)/3168.
%F A351770 G.f.: x*(1 + 1044*x + 54462*x^2 + 595860*x^3 + 2048388*x^4 + 2563644*x^5 + 1193226*x^6 + 188508*x^7 + 7635*x^8 + 32*x^9)/(1-x)^13. - _Robert Israel_, Feb 18 2022
%p A351770 seq(n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168,
%p A351770 n=0..30);# _Robert Israel_, Feb 18 2022
%o A351770 (PARI) {a(n) = n*(n+1)*(n+2)*(44*n^9+276*n^8+492*n^7-48*n^6-609*n^5+207*n^4+487*n^3-291*n^2-90*n+60)/3168};
%o A351770 (PARI) a(n) = sum(j=1, n, sum(i=1, j, i^5*j^5));
%Y A351770 Cf. A001296 (for power 1), A060493 (for squares), A346642 (for cubes), A351766 (for fourth powers).
%Y A351770 Cf. A000584 (fifth powers), A000539 (sum of fifth powers).
%K A351770 nonn,easy
%O A351770 0,3
%A A351770 _Roudy El Haddad_, Feb 18 2022