%I #16 Sep 08 2022 08:45:29
%S 580,1175,2070,3325,5000,7155,9850,13145,17100,21775,27230,33525,
%T 40720,48875,58050,68305,79700,92295,106150,121325,137880,155875,
%U 175370,196425,219100,243455,269550,297445,327200,358875,392530,428225,466020
%N Absolute value of coefficient of x^3 in polynomial whose zeros are 5 consecutive integers starting with the n-th integer.
%C Sums of all distinct products of 3 out of 5 consecutive integers, starting with the n-th integer; value of 3rd elementary symmetric function on the 5 consecutive integers. cf. Vieta's formulas.
%H Vincenzo Librandi, <a href="/A127694/b127694.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 5*(n+3)*(2*n^2+12*n+15). G.f.: 5*x*(116-229*x+170*x^2-45*x^3)/(1-x)^4. [Colin Barker, Mar 28 2012]
%F a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Vincenzo Librandi_, Jun 28 2012
%t r = {}; k = 0; a = {}; Do[Do[Do[If[(d != b) && (d != c) && (b != c), AppendTo[a, {d, b, c}]], {c, b, 5}], {b, d, 5}], {d, 1, 5}]; Do[Do[k = k + Sum[(x + a[[v, 1]]) (x + a[[v, 2]]) (x + a[[v, 3]]), {v, 1, Length[a]}]]; AppendTo[r, k]; k = 0, {x, 1, 50}]; r
%t CoefficientList[Series[5*(116-229*x+170*x^2-45*x^3)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 28 2012 *)
%o (Magma) I:=[580, 1175, 2070, 3325]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Jun 28 2012
%Y Cf. A127694, A127350.
%K nonn,easy
%O 1,1
%A _Artur Jasinski_, Jan 23 2007