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A000596
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Central factorial numbers.
(Formerly M3686 N1505)
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2
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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
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OFFSET
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3,1
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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John Cerkan, Table of n, a(n) for n = 3..10000
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.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for sequences related to factorial numbers
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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Column 2 of triangle A008955.
Sequence in context: A078187 A100256 A163944 * A113525 A290263 A224538
Adjacent sequences: A000593 A000594 A000595 * A000597 A000598 A000599
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Minor edits by Vaclav Kotesovec, Feb 23 2015
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STATUS
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approved
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