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A000596 Central factorial numbers.
(Formerly M3686 N1505)
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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 *)

CROSSREFS

Column 2 of triangle A008955.

Sequence in context: A078187 A100256 A163944 * A113525 A290263 A224538

Adjacent sequences:  A000593 A000594 A000595 * A000597 A000598 A000599

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Minor edits by Vaclav Kotesovec, Feb 23 2015

STATUS

approved

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Last modified April 21 23:52 EDT 2021. Contains 343156 sequences. (Running on oeis4.)