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 A052787 A simple grammar. Product of 5 consecutive integers. 14

%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). - _Reinhard Zumkeller_, Feb 19 2010

%C 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

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

%H Thomas Harriot, <a href="http://echo.mpiwg-berlin.mpg.de/MPIWG:5PYT50NY">Manuscript 6782</a>, <a href="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView?url=%2Fmpiwg%2Fonline%2Fpermanent%2Flibrary%2FHSPGZ0AE%2F&amp;viewMode=images&amp;tocMode=thumbs&amp;tocPN=1&amp;query=&amp;searchPN=1&amp;queryType=&amp;characterNormalization=reg&amp;pn=153">p. 77</a>, c. 1599.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=744">Encyclopedia of Combinatorial Structures 744</a>

%H <a href="/index/Rec#order_06">Index entries for 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 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

%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 DelĂ©ham_, Dec 12 2003

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