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A052787 A simple grammar. Product of 5 consecutive integers. 13

%I

%S 0,0,0,0,0,120,720,2520,6720,15120,30240,55440,95040,154440,240240,

%T 360360,524160,742560,1028160,1395360,1860480,2441880,3160080,4037880,

%U 5100480,6375600,7893600,9687600,11793600,14250600,17100720,20389320

%N A simple grammar. Product of 5 consecutive integers.

%C For n>5: a(n) = A173333(n,n-5). [From _Reinhard Zumkeller_, Feb 19 2010]

%H Vincenzo Librandi, <a href="/A052787/b052787.txt">Table of n, a(n) for n = 0..1000</a>

%H INRIA Algorithms Project, <a href="http://algo.inria.fr/ecs/ecs?searchType=1&amp;service=Search&amp;searchTerms=744">Encyclopedia of Combinatorial Structures 744</a>

%H <a href="/index/Rea#recLCC">Index to sequences with linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).

%F a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)=n!/(n-5)!.

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

%F 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}

%F O.g.f.: 120*x^5/(-1+x)^6. - _R. J. Mathar_, Nov 16 2007

%F a(n) = a(n-1) + 5*A052762(n). - _J. M. Bergot_, May 30 2012

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

%p seq(numbperm (n,5), n=0..31); - _Zerinvary Lajos_, Apr 26 2007

%p restart: 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);# [From _Zerinvary Lajos_, Apr 05 2009]

%t Times@@@(Partition[Range[-4,35],5,1]) (* _Harvey P. Dale_, Feb 04 2011 *)

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

%o (PARI) a(n)=120*binomial(n,5) \\ _Charles R Greathouse IV_, Nov 20 2011

%Y Cf. A002378, A007531, A052762.

%Y Equals 120 * C(n, 5) = 120 * A000389(n).

%Y Equals 4 * A054559.

%K easy,nonn

%O 0,6

%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

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Last modified September 17 07:07 EDT 2014. Contains 246836 sequences.