%I M3686 N1505 #101 Dec 07 2023 04:03:08

%S 4,49,273,1023,3003,7462,16422,32946,61446,108031,180895,290745,

%T 451269,679644,997084,1429428,2007768,2769117,3757117,5022787,6625311,

%U 8632866,11123490,14185990,17920890,22441419,27874539,34362013,42061513,51147768,61813752,74271912

%N Central factorial numbers.

%C 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 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H John Cerkan, <a href="/A000596/b000596.txt">Table of n, a(n) for n = 3..10000</a>

%H Roudy El Haddad, <a href="https://arxiv.org/abs/2102.00821">Multiple Sums and Partition Identities</a>, arXiv:2102.00821 [math.CO], 2021.

%H 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 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 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 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (1/360)*n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(5*n+1).

%F a(n+1/2) = (1/16)*A001823(n).

%F 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 From _Roudy El Haddad_, Feb 17 2022: (Start)

%F a(n) = Sum_{0 < i < j < n} (i*j)^2.

%F a(n) = binomial(2n,5)*(5*n+1)/4!. (End)

%p A000596:=-(4+21*z+14*z**2+z**3)/(z-1)**7; # conjectured by _Simon Plouffe_ in his 1992 dissertation

%p 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 f[k_] := k^2; t[n_] := Table[f[k], {k, 1, n}]

%t a[n_] := SymmetricPolynomial[2, t[n]]

%t Table[a[n], {n, 2, 32}] (* A000596 *)

%t (* _Clark Kimberling_, Dec 31 2011 *)

%t 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 (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 Column 2 of triangle A008955.

%Y Cf. A000290 (squares), A000330 (sum of squares), A000597 (order 3).

%K nonn,easy

%O 3,1

%A _N. J. A. Sloane_

%E Minor edits by _Vaclav Kotesovec_, Feb 23 2015