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A159083 Products of 7 consecutive integers.

%I A159083 #45 Mar 09 2022 10:06:41
%S A159083 0,0,0,0,0,0,0,5040,40320,181440,604800,1663200,3991680,8648640,
%T A159083 17297280,32432400,57657600,98017920,160392960,253955520,390700800,
%U A159083 586051200,859541760,1235591280,1744364160,2422728000,3315312000,4475671200,5967561600,7866331200
%N A159083 Products of 7 consecutive integers.
%H A159083 Michael De Vlieger and Harvey P. Dale, <a href="/A159083/b159083.txt">Table of n, a(n) for n = 0..10000</a> (first 1000 terms by Harvey P. Dale.)
%H A159083 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A159083 E.g.f.: x^7*exp(x).
%F A159083 For n>=8: a(n) = A173333(n,n-7). - _Reinhard Zumkeller_, Feb 19 2010
%F A159083 G.f.: 5040*x^7/(1-x)^8. - _Colin Barker_, Mar 27 2012
%F A159083 From _Amiram Eldar_, Mar 08 2022: (Start)
%F A159083 a(n) = n*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6) = n!/(n-7)!.
%F A159083 Sum_{n>=7} 1/a(n) = 1/4320.
%F A159083 Sum_{n>=7} (-1)^(n+1)/a(n) = 4*log(2)/45 - 1327/21600. (End)
%p A159083 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);
%t A159083 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 *)
%o A159083 (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
%o A159083 (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
%Y A159083 Cf. A052762, A052787, A053625, A173333.
%Y A159083 Equals A008279(n,7) (for n>=7).
%K A159083 nonn,easy
%O A159083 0,8
%A A159083 _Zerinvary Lajos_, Apr 05 2009