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URL: https://oeis.org/A000596

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A000596
Central factorial numbers: A008955(n,2).
7
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
3,1
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
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
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)
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 *)
PROG
(PARI) {a(n) = n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(5*n+1)/360}; \\ Roudy El Haddad, Feb 17 2022
CROSSREFS
Column 2 of triangle A008955.
Cf. A000290 (squares), A000330 (sum of squares), A000597 (order 3).
Sequence in context: A078187 A100256 A163944 * A113525 A290263 A224538
KEYWORD
nonn,easy
EXTENSIONS
Minor edits by Vaclav Kotesovec, Feb 23 2015
STATUS
approved