%I #109 Aug 28 2024 10:26:36

%S 0,0,0,0,24,120,360,840,1680,3024,5040,7920,11880,17160,24024,32760,

%T 43680,57120,73440,93024,116280,143640,175560,212520,255024,303600,

%U 358800,421200,491400,570024,657720,755160,863040,982080,1113024

%N Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).

%C 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

%C Number of n-colorings of the complete graph on 4 vertices, which is also the tetrahedral graph. - _Eric M. Schmidt_, Oct 31 2012

%C Cf. A130534 for relations to colored forests and disposition of flags on flagpoles. - _Tom Copeland_, Apr 05 2014

%C Number of 4-permutations of the set {1, 2, ..., n}. - _Joerg Arndt_, Apr 05 2014

%H Eric M. Schmidt, <a href="/A052762/b052762.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=719">Encyclopedia of Combinatorial Structures 719</a>

%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv:1406.3081 [math.CO], 2014.

%H Michelle Rudolph-Lilith, <a href="http://arxiv.org/abs/1508.07894">On the Product Representation of Number Sequences, with Application to the Fibonacci Family</a>, arXiv:1508.07894 [math.NT], 2015.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclicQuadrilateral.html">Cyclic Quadrilateral</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Pochhammer_symbol">Pochhammer symbol</a>.

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(n) = n*(n-1)*(n-2)*(n-3) = n!/(n-4)! (for n >= 4).

%F a(n) = A001094(n) - n.

%F E.g.f.: x^4*exp(x).

%F Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-1-n)*a(n) + (n-3)*a(n+1)}.

%F a(n) + 1 = A062938(n-4) for n > 4. - _Amarnath Murthy_, Dec 13 2003

%F a(n) = numbperm(n, 4). - _Zerinvary Lajos_, Apr 26 2007

%F O.g.f.: -24*x^4/(-1+x)^5. - _R. J. Mathar_, Nov 23 2007

%F For n > 4: a(n) = A173333(n, n-4). - _Reinhard Zumkeller_, Feb 19 2010

%F 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

%F a(n) = a(n-1) + 4*A007531(n). - _J. M. Bergot_, May 30 2012

%F 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

%F a(n) - 1 = A069756(n-2) for n >= 4. - _Jean-Christophe Hervé_, Nov 01 2015

%F a(n) = 24 * A000332(n). - _Bruce J. Nicholson_, Apr 03 2017

%F From _R. J. Mathar_, Jun 30 2021: (Start)

%F Sum_{n>=4} 24*(-1)^n/a(n) = A242023.

%F Sum_{n>=4} 1/a(n) = 1/18. (End)

%p spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

%p seq(numbperm (n,4), n=0..34); # _Zerinvary Lajos_, Apr 26 2007

%p 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

%t Table[n*(n+1)*(n+2)*(n+3), {n,-3,60}] (* _Vladimir Joseph Stephan Orlovsky_, Apr 21 2010 *)

%t Times@@@Partition[Range[-3,60], 4, 1] (* _Harvey P. Dale_, May 09 2012 *)

%t LinearRecurrence[ {5,-10,10,-5,1}, {0,0,0,0,24}, 60] (* _Harvey P. Dale_, May 09 2012 *)

%o (PARI) a(n)=24*binomial(n,4) \\ _Charles R Greathouse IV_, Nov 20 2011

%o (Maxima) A052762(n):=n*(n-1)*(n-2)*(n-3)$

%o makelist(A052762(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */

%o (Magma) [n*(n-1)*(n-2)*(n-3): n in [0..30]]; // _G. C. Greubel_, Nov 19 2017

%Y Cf. A001094, A002378, A007531, A052787.

%K easy,nonn

%O 0,5

%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000

%E More terms from _Henry Bottomley_, Mar 20 2000

%E Formula corrected by _Philippe Deléham_, Dec 12 2003