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 A000597 M5255 N2287 #108 Jan 13 2025 10:46:26
%S A000597 36,820,7645,44473,191620,669188,1999370,5293970,12728936,28285400,
%T A000597 58856655,115842675,217378200,391367064,679524340,1142659012,
%U A000597 1867463260,2975110060,4631998657,7063027565,10567817084,15540347900,22492529150,32082258390,45146587200
%N A000597 Central factorial numbers: A008955(n,3).
%C A000597 a(n) is the sum of all products of three distinct squares of positive integers up to n, i.e., the sum of all products of three distinct elements from the set of squares {1^2, ..., (n-1)^2}. - _Roudy El Haddad_, Feb 17 2022
%D A000597 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A000597 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000597 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000597 Vincenzo Librandi, <a href="/A000597/b000597.txt">Table of n, a(n) for n = 4..1000</a>
%H A000597 Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.
%H A000597 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. (See Example 5.2 and Theorem 5.1)
%H A000597 Mircea Merca, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL15/Merca2/merca7.html">A Special Case of the Generalized Girard-Waring Formula</a>, J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
%H A000597 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H A000597 <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 A000597 O.g.f.: x^4 * (x^5 + 75*x^4 + 603*x^3 + 1065*x^2 + 460*x + 36) / (1-x)^10.
%F A000597 a(n) = s(n,n-3)^2-2*s(n,n-4)*s(n,n-2)+2*s(n,n-5)*s(n,n-1)+2*s(n,n-6), where s(n,k) are Stirling numbers of the first kind, A048994. - _Mircea Merca_, Apr 03 2012
%F A000597 From _Roudy El Haddad_, Feb 17 2022: (Start)
%F A000597 a(n) = Sum_{0 < i < j < k < n} (i*j*k)^2.
%F A000597 a(n) = (n - 1)*(n - 2)*(n - 3)*n*(2*n-1)*(2*n - 3)*(2*n - 5)*(35*n^2 + 21*n + 4)/45360.
%F A000597 a(n) = (1/(9!*2))*((2*n)!/(2*n-7)!)*(35*n^2 + 21*n + 4).
%F A000597 a(n) = binomial(2*n,7)*(35*n^2 + 21*n + 4)/144. (End)
%p A000597 1/(-1+z)^10*(z^5+75*z^4+603*z^3+1065*z^2+460*z+36);
%p A000597 seq(stirling1(n,n-3)^2-2*stirling1(n,n-4)*stirling1(n,n-2)+2*stirling1(n,n-5)*stirling1(n,n-1)+2*stirling1(n,n-6),n=0..30); # _Mircea Merca_, Apr 03 2012
%t A000597 CoefficientList[Series[(x^5 + 75*x^4 + 603*x^3 + 1065*x^2 + 460*x + 36)/(1-x)^10, {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 23 2015 *)
%t A000597 LinearRecurrence[{10, -45, 120, -210, 252, -210, 120, -45, 10, -1}, {36, 820, 7645, 44473, 191620, 669188, 1999370, 5293970, 12728936, 28285400}, 40] (* _Vincenzo Librandi_, Aug 07 2017 *)
%o A000597 (PARI) {a(n) = (n-1)*(n-2)*(n-3)*(n)*(2*n-1)*(2*n-3)*(2*n-5)*(35*n^2+21*n+4)/45360}; \\ _Roudy El Haddad_, Feb 17 2022
%Y A000597 Column 3 of triangle A008955.
%Y A000597 Cf. A000290 (squares), A000330 (sum of squares), A000596 (for two squares).
%Y A000597 Cf. A001303 (for power 1).
%K A000597 nonn,easy
%O A000597 4,1
%A A000597 _N. J. A. Sloane_