A000596 Central factorial numbers: A008955(n,2).

4, 49, 273, 1023, 3003, 7462, 16422, 32946, 61446, 108031, 180895, 290745, 451269, 679644, 997084, 1429428, 2007768, 2769117, 3757117, 5022787, 6625311, 8632866, 11123490, 14185990, 17920890, 22441419, 27874539, 34362013, 42061513, 51147768, 61813752, 74271912

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

References

• J. Riordan, Combinatorial Identities, Wiley, 1968, p. 217.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• John Cerkan, Table of n, a(n) for n = 3..10000
• Roudy El Haddad, Multiple Sums and Partition Identities, arXiv:2102.00821 [math.CO], 2021.
• Roudy El Haddad, A generalization of multiple zeta value. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 2022, 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.
• Mircea Merca, A Special Case of the Generalized Girard-Waring Formula J. Integer Sequences, Vol. 15 (2012), Article 12.5.7.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Index entries for sequences related to factorial numbers
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Column 2 of triangle A008955.

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

Programs

Maple

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

Mathematica

f[k_] := k^2; t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[2, t[n]]
Table[a[n], {n, 2, 32}]  (* A000596 *)
(* Clark Kimberling, Dec 31 2011 *)
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 *)

PARI

{a(n) = n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(5*n+1)/360}; \\ Roudy El Haddad, Feb 17 2022

Formula

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

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

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

From Roudy El Haddad, Feb 17 2022: (Start)

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

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

Extensions

Minor edits by Vaclav Kotesovec, Feb 23 2015