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, 505975, 548150, 592605

Offset: 1

Author

Artur Jasinski, Jan 23 2007

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A127694, A127350.

Programs

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

Mathematica

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

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