A052762 Products of 4 consecutive integers: a(n) = n*(n-1)*(n-2)*(n-3).

0, 0, 0, 0, 24, 120, 360, 840, 1680, 3024, 5040, 7920, 11880, 17160, 24024, 32760, 43680, 57120, 73440, 93024, 116280, 143640, 175560, 212520, 255024, 303600, 358800, 421200, 491400, 570024, 657720, 755160, 863040, 982080, 1113024

Offset: 0

Author

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

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A001094, A002378, A007531, A052787.

Programs

Magma

[n*(n-1)*(n-2)*(n-3): n in [0..30]]; // G. C. Greubel, Nov 19 2017

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 *)

Maxima

A052762(n):=n*(n-1)*(n-2)*(n-3)$
makelist(A052762(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

PARI

a(n)=24*binomial(n,4) \\ Charles R Greathouse IV, Nov 20 2011

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)

Extensions

More terms from Henry Bottomley, Mar 20 2000

Formula corrected by Philippe Deléham, Dec 12 2003