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]
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
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from Henry Bottomley, Mar 20 2000
STATUS
approved