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 A000596 M3686 N1505 #104 Jan 13 2025 10:34:04
%S A000596 4,49,273,1023,3003,7462,16422,32946,61446,108031,180895,290745,
%T A000596 451269,679644,997084,1429428,2007768,2769117,3757117,5022787,6625311,
%U A000596 8632866,11123490,14185990,17920890,22441419,27874539,34362013,42061513,51147768,61813752,74271912
%N A000596 Central factorial numbers: A008955(n,2).
%C A000596 a(n) is the sum of the products of each unique pair of elements of the set {1, 4, 9, 16, ... , (n-1)^2} (a(3) = 1*4, a(4) = 1*4 + 1*9 + 4*9, a(5) = 1*4 + 1*9 + 1*16 + 4*9 + 4*16 + 9*16, etc.) - _Jeffreylee R. Snow_, Sep 23 2013
%D A000596 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
%D A000596 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000596 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000596 John Cerkan, <a href="/A000596/b000596.txt">Table of n, a(n) for n = 3..10000</a>
%H A000596 Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.
%H A000596 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 A000596 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 A000596 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A000596 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H A000596 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H A000596 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A000596 a(n) = (1/360)*n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(5*n+1).
%F A000596 a(n+1/2) = (1/16)*A001823(n).
%F A000596 a(n) = s(n,n-2)^2-2*s(n,n-3)*s(n,n-1)+2*s(n,n-4), where s(n,k) are Stirling numbers of the first kind, A048994. - _Mircea Merca_, Apr 03 2012
%F A000596 From _Roudy El Haddad_, Feb 17 2022: (Start)
%F A000596 a(n) = Sum_{0 < i < j < n} (i*j)^2.
%F A000596 a(n) = binomial(2n,5)*(5*n+1)/4!. (End)
%p A000596 A000596:=-(4+21*z+14*z**2+z**3)/(z-1)**7; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p A000596 seq(stirling1(n,n-2)^2-2*stirling1(n,n-3)*stirling1(n,n-1)+2*stirling1(n,n-4),n=0..50); # _Mircea Merca_, Apr 03 2012
%t A000596 f[k_] := k^2; t[n_] := Table[f[k], {k, 1, n}]
%t A000596 a[n_] := SymmetricPolynomial[2, t[n]]
%t A000596 Table[a[n], {n, 2, 32}] (* A000596 *)
%t A000596 (* _Clark Kimberling_, Dec 31 2011 *)
%t A000596 a[n_] := 1/360 * n * (n - 1) * (n - 2) * (2n - 1) * (2n - 3) * (5n + 1);Table[a[n],{n,3,34}] (* _James C. McMahon_, Dec 05 2023 *)
%o A000596 (PARI) {a(n) = n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(5*n+1)/360}; \\ _Roudy El Haddad_, Feb 17 2022
%Y A000596 Column 2 of triangle A008955.
%Y A000596 Cf. A000290 (squares), A000330 (sum of squares), A000597 (order 3).
%K A000596 nonn,easy
%O A000596 3,1
%A A000596 _N. J. A. Sloane_
%E A000596 Minor edits by _Vaclav Kotesovec_, Feb 23 2015