A052787 Product of 5 consecutive integers.

0, 0, 0, 0, 0, 120, 720, 2520, 6720, 15120, 30240, 55440, 95040, 154440, 240240, 360360, 524160, 742560, 1028160, 1395360, 1860480, 2441880, 3160080, 4037880, 5100480, 6375600, 7893600, 9687600, 11793600, 14250600, 17100720, 20389320, 24165120, 28480320, 33390720

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

Appears in Harriot along with the formula (for a different offset) a(n) = n^5 + 10n^4 + 35n^3 + 50n^2 + 24n, see links. - Charles R Greathouse IV, Oct 22 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Thomas Harriot, Manuscript 6782, p. 77, c. 1599.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 744.
• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A002378, A007531, A052762, A000389, A054559, A173333.

Programs

Magma

[n*(n-1)*(n-2)*(n-3)*(n-4): n in [0..35]]; // Vincenzo Librandi, May 26 2011

Maple

spec := [S,{B=Set(Z),S=Prod(Z,Z,Z,Z,Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(numbperm (n,5), n=0..31); # Zerinvary Lajos, Apr 26 2007
G(x):=x^5*exp(x): f[0]:=G(x): for n from 1 to 31 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..31); # Zerinvary Lajos, Apr 05 2009

Mathematica

Times@@@(Partition[Range[-4,35],5,1])  (* Harvey P. Dale, Feb 04 2011 *)

PARI

a(n)=120*binomial(n,5) \\ Charles R Greathouse IV, Nov 20 2011

Formula

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)=n!/(n-5)!. [Corrected by Philippe Deléham, Dec 12 2003]

a(n) = 120*A000389(n) = 4*A054559(n).

E.g.f.: x^5*exp(x).

Recurrence: {a(1)=0, a(2)=0, a(4)=0, a(3)=0, (-1-n)*a(n)+(-4+n)*a(n+1), a(5)=120}.

O.g.f.: 120*x^5/(-1+x)^6. - R. J. Mathar, Nov 16 2007

For n>5: a(n) = A173333(n,n-5). - Reinhard Zumkeller, Feb 19 2010

a(n) = a(n-1) + 5*A052762(n). - J. M. Bergot, May 30 2012

From Amiram Eldar, Mar 08 2022: (Start)

Sum_{n>=5} 1/a(n) = 1/96.

Sum_{n>=5} (-1)^(n+1)/a(n) = 2*log(2)/3 - 131/288. (End)

Extensions

More terms from Henry Bottomley, Mar 20 2000