OFFSET
0,5
COMMENTS
Also, starting with n=4, the square of area of cyclic quadrilateral with sides n, n-1, n-2, n-3. - Zak Seidov, Jun 20 2003
Number of n-colorings of the complete graph on 4 vertices, which is also the tetrahedral graph. - Eric M. Schmidt, Oct 31 2012
Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - Tom Copeland, Apr 05 2014
Number of 4-permutations of the set {1, 2, ..., n}. - Joerg Arndt, Apr 05 2014
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 719
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv:1406.3081 [math.CO], 2014.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv:1508.07894 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cyclic Quadrilateral
Wikipedia, Pochhammer symbol.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
a(n) = n*(n-1)*(n-2)*(n-3) = n!/(n-4)! (for n >= 4).
a(n) = A001094(n) - n.
E.g.f.: x^4*exp(x).
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-1-n)*a(n) + (n-3)*a(n+1)}.
a(n) + 1 = A062938(n-4) for n > 4. - Amarnath Murthy, Dec 13 2003
a(n) = numbperm(n, 4). - Zerinvary Lajos, Apr 26 2007
O.g.f.: -24*x^4/(-1+x)^5. - R. J. Mathar, Nov 23 2007
For n > 4: a(n) = A173333(n, n-4). - Reinhard Zumkeller, Feb 19 2010
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5), with a(0)=0, a(1)=0, a(2)=0, a(3)=0, a(4)=24. - Harvey P. Dale, May 09 2012
a(n) = a(n-1) + 4*A007531(n). - J. M. Bergot, May 30 2012
a(n) = (n)_4 = Pochhammer(n,4), using the "falling factorial" convention; other authors write Pochhammer(x,k) for what is denoted x^(k) in the Wikipedia article, then a(n) = (n-3)^(4). - M. F. Hasler, Oct 20 2013
a(n) - 1 = A069756(n-2) for n >= 4. - Jean-Christophe Hervé, Nov 01 2015
a(n) = 24 * A000332(n). - Bruce J. Nicholson, Apr 03 2017
From R. J. Mathar, Jun 30 2021: (Start)
Sum_{n>=4} 24*(-1)^n/a(n) = A242023.
Sum_{n>=4} 1/a(n) = 1/18. (End)
MAPLE
spec := [S, {B=Set(Z), S=Prod(Z, Z, Z, Z, B)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(numbperm (n, 4), n=0..34); # Zerinvary Lajos, Apr 26 2007
G(x):=x^4*exp(x): f[0]:=G(x): for n from 1 to 34 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..34); # Zerinvary Lajos, Apr 05 2009
MATHEMATICA
Table[n*(n+1)*(n+2)*(n+3), {n, -3, 60}] (* Vladimir Joseph Stephan Orlovsky, Apr 21 2010 *)
Times@@@Partition[Range[-3, 60], 4, 1] (* Harvey P. Dale, May 09 2012 *)
LinearRecurrence[ {5, -10, 10, -5, 1}, {0, 0, 0, 0, 24}, 60] (* Harvey P. Dale, May 09 2012 *)
PROG
(PARI) a(n)=24*binomial(n, 4) \\ Charles R Greathouse IV, Nov 20 2011
(Maxima) A052762(n):=n*(n-1)*(n-2)*(n-3)$
makelist(A052762(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
(Magma) [n*(n-1)*(n-2)*(n-3): n in [0..30]]; // G. C. Greubel, Nov 19 2017
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
More terms from Henry Bottomley, Mar 20 2000
Formula corrected by Philippe Deléham, Dec 12 2003
STATUS
approved