A159083 Products of 7 consecutive integers.

0, 0, 0, 0, 0, 0, 0, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 17297280, 32432400, 57657600, 98017920, 160392960, 253955520, 390700800, 586051200, 859541760, 1235591280, 1744364160, 2422728000, 3315312000, 4475671200, 5967561600, 7866331200

Offset

0,8

Links

Michael De Vlieger and Harvey P. Dale, Table of n, a(n) for n = 0..10000 (first 1000 terms by Harvey P. Dale.)

Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Formula

E.g.f.: x^7*exp(x).

For n>=8: a(n) = A173333(n,n-7). - Reinhard Zumkeller, Feb 19 2010

G.f.: 5040*x^7/(1-x)^8. - Colin Barker, Mar 27 2012

From Amiram Eldar, Mar 08 2022: (Start)

a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6) = n!/(n-7)!.

Sum_{n>=7} 1/a(n) = 1/4320.

Sum_{n>=7} (-1)^(n+1)/a(n) = 4*log(2)/45 - 1327/21600. (End)

Maple

G(x):=x^7*exp(x): f[0]:=G(x): for n from 1 to 36 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..33);

Mathematica

Table[Times@@(n+Range[0, 6]), {n, -6, 25}] (* or *) LinearRecurrence[{8, -28, 56, -70, 56, -28, 8, -1}, {0, 0, 0, 0, 0, 0, 0, 5040}, 30] (* Harvey P. Dale, Apr 07 2018 *)

Prog

(PARI) my(x='x+O('x^30)); concat([0, 0, 0, 0, 0, 0, 0], Vec(5040*x^7/(1-x)^8)) \\ G. C. Greubel, Jun 28 2018
(Magma) I:=[0, 0, 0, 0, 0, 0, 0, 5040]; [n le 8 select I[n] else 8*Self(n-1) - 28*Self(n-2) +56*Self(n-3) -70*Self(n-4) +56*Self(n-5) -28*Self(n-6) +8*Self(n-7) -Self(n-8): n in [1..30]]; // G. C. Greubel, Jun 28 2018

Crossrefs

Cf. A052762, A052787, A053625, A173333.

Equals A008279(n,7) (for n>=7).

Sequence in context: A321843 A226886 A284204 * A179731 A061140 A061122

Adjacent sequences: A159080 A159081 A159082 * A159084 A159085 A159086

Keyword

nonn,easy

Author

Zerinvary Lajos, Apr 05 2009

Status

approved