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
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).
Programs
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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
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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);
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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 *) -
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
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)