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 A351105 #49 Apr 02 2025 15:57:27
%S A351105 0,1,85,1408,11440,61490,251498,846260,2458676,6369275,15047175,
%T A351105 32955780,67746900,131969604,245444980,438485080,756163672,1263878005,
%U A351105 2054474617,3257248280,5049161480,7668672374,11432601950,16756516140,24179145900,34391417775
%N A351105 a(n) = Sum_{k=1..n} Sum_{j=1..k} Sum_{i=1..j} (i*j*k)^2.
%C A351105 a(n) is the sum of all products of three squares of positive integers up to n, i.e., the sum of all products of three elements from the set of squares {1^2, ..., n^2}.
%H A351105 Harvey P. Dale, <a href="/A351105/b351105.txt">Table of n, a(n) for n = 0..1000</a>
%H A351105 Roudy El Haddad, <a href="https://arxiv.org/abs/2101.09089">Recurrent Sums and Partition Identities</a>, arXiv:2101.09089 [math.NT], 2021.
%H A351105 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 A351105 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A351105 a(n) = n*(n+1)*(n+2)*(n+3)*(2n+1)*(2n+3)*(2n+5)*(35*n^2-21*n+4)/45360 (from the recurrent form of Faulhaber's formula).
%F A351105 a(n) = (1/(9!*2))*((2n+6)!/(2n-1)!)*(35*n^2-21*n+4).
%F A351105 a(n) = binomial(2n+6,7)*(35*n^2-21*n+4)/144.
%F A351105 G.f.: x*(36*x^5+460*x^4+1065*x^3+603*x^2+75*x+1)/(x-1)^10. - _Alois P. Heinz_, Jan 31 2022
%F A351105 a(n) = 2/(2*n)! * Sum_{j = 1..n} (-1)^(n+j) * binomial(2*n, n-j) * j^(2*n+6). - _Peter Bala_, Mar 31 2025
%t A351105 CoefficientList[Series[x (36 x^5 + 460 x^4 + 1065 x^3 + 603 x^2 + 75 x + 1)/(x - 1)^10, {x, 0, 25}], x] (* _Michael De Vlieger_, Feb 04 2022 *)
%t A351105 LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,85,1408,11440,61490,251498,846260,2458676,6369275},30] (* _Harvey P. Dale_, Jul 18 2022 *)
%o A351105 (PARI) {a(n) = n*(n+1)*(n+2)*(n+3)*(2*n+1)*(2*n+3)*(2*n+5)*(35*n^2-21*n+4)/45360};
%o A351105 (PARI) a(n) = sum(i=1, n, sum(j=1, i, sum(k=1, j, i^2*j^2*k^2)));
%o A351105 (Python)
%o A351105 def A351105(n): return n*(n*(n*(n*(n*(n*(n*(n*(280*n + 2772) + 10518) + 18711) + 14385) + 1323) - 2863) - 126) + 360)//45360 # _Chai Wah Wu_, Feb 17 2022
%Y A351105 A diagonal of A036969.
%Y A351105 Cf. A000290 (squares), A000330 (sum of squares), A060493 (for two squares).
%Y A351105 Cf. A001297 (for power 1).
%K A351105 nonn,easy
%O A351105 0,3
%A A351105 _Roudy El Haddad_, Jan 31 2022