A127694 Absolute value of coefficient of x^3 in polynomial whose zeros are 5 consecutive integers starting with the n-th integer.

580, 1175, 2070, 3325, 5000, 7155, 9850, 13145, 17100, 21775, 27230, 33525, 40720, 48875, 58050, 68305, 79700, 92295, 106150, 121325, 137880, 155875, 175370, 196425, 219100, 243455, 269550, 297445, 327200, 358875, 392530, 428225, 466020

Offset

1,1

Comments

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.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

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]

a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - Vincenzo Librandi, Jun 28 2012

Mathematica

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
CoefficientList[Series[5*(116-229*x+170*x^2-45*x^3)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2012 *)

Prog

(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

Crossrefs

Cf. A127694, A127350.

Sequence in context: A097769 A186786 A185609 * A185605 A186784 A185606

Adjacent sequences: A127691 A127692 A127693 * A127695 A127696 A127697

Keyword

nonn,easy

Author

Artur Jasinski, Jan 23 2007

Status

approved