A231146 Array of coefficients of numerator polynomials of the rational function p(n, x - 1/x), where p(n,x) is the n-th cyclotomic polynomial.

1, -1, -1, 1, -1, 1, 1, 1, -1, -1, 1, 1, 1, 0, -1, 0, 1, 1, -1, -3, 2, 5, -2, -3, 1, 1, 1, 1, -1, -1, 1, 1, -1, -5, 4, 12, -8, -15, 8, 12, -4, -5, 1, 1, 1, 0, -4, 0, 7, 0, -4, 0, 1, 1, 0, -6, -1, 15, 3, -19, -3, 15, 1, -6, 0, 1, 1, 1, -3, -2, 5, 2, -3, -1, 1

Offset

0,20

Comments

Define q(n,x) = p(n, x - 1/x). If r is a zero of p(n,x) then (1/2)*(r +- sqrt(r^2 + 4)) are zeros of q(n,x).

Links

Table of n, a(n) for n=0..74.

Example

First 6 rows:
   1
  -1, -1,  1
  -1,  1,  1
   1, -1, -1,  1,  1
   1,  0, -1,  0,  1
   1, -1, -3,  2,  5, -2, -3,  1,  1
First 4 polynomials:
   1,
  -1 - x + x^2,
  -1 + x + x^2,
   1 - x - x^2 + x^3 + x^4.

Mathematica

z = 60; p[n_, x_] := p[x] = Cyclotomic[n, x]; Table[p[n, x], {n, 0, z/4}]; f1[n_, x_] := f1[n, x] = Numerator[Factor[p[n, x] /. x -> x - 1/x]]; Table[Expand[f1[n, x]], {n, 0, z/4}]
t = Flatten[Table[CoefficientList[f1[n, x], x], {n, 0, z/4}]]

Crossrefs

Cf. A230003.

Sequence in context: A074830 A369043 A182983 * A287692 A208889 A127750

Adjacent sequences: A231143 A231144 A231145 * A231147 A231148 A231149

Keyword

tabf,sign,easy

Author

Clark Kimberling, Nov 07 2013

Status

approved