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

0,6

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).

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)

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 *)

Prog

(Magma) [n*(n-1)*(n-2)*(n-3)*(n-4): n in [0..35]]; // Vincenzo Librandi, May 26 2011
(PARI) a(n)=120*binomial(n, 5) \\ Charles R Greathouse IV, Nov 20 2011

Crossrefs

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

Sequence in context: A005820 A306602 A300299 * A292970 A052769 A179724

Adjacent sequences: A052784 A052785 A052786 * A052788 A052789 A052790

Keyword

easy,nonn

Author

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

Extensions

More terms from Henry Bottomley, Mar 20 2000

Status

approved