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A230002 Array of coefficients of numerator polynomials of the rational function p(n, x - 1/x), where p(n,x) is the Fibonacci polynomial defined by p(1,x) = 1, p(2,x) = x, p(n,x) = x*p(n-1,x) + p(n-2,x). 2
1, -1, 0, 1, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0

COMMENTS

Row n has 2n-1 terms.  If r is a zero of p(n,x) then (1/2)(r +- sqrt(r^2 + 4) are zeros of q(n,x).  Appears to be a signed version of A071028.

LINKS

Table of n, a(n) for n=0..79.

EXAMPLE

First 5 rows: (1}, (-1,0,1), (1,0,-1,0,1), (-1,0,1,0,-1,0,1).

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

MATHEMATICA

p[n_, x_] := p[x] = Fibonacci[n, x]; Table[p[n, x], {n, 1, 10}]

f[n_, x_] := f[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1/x]]]

g[n_, x_] := g[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x - 1/x]]]

h[n_, x_] := h[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1 + 1/x]]]

t1 = Flatten[Table[CoefficientList[f[n, x], x], {n, 1, 12}]];  (* A229995 *)

t2 = Flatten[Table[CoefficientList[g[n, x], x], {n, 1, 12}]];  (* A230002 *)

t3 = Flatten[Table[CoefficientList[h[n, x], x], {n, 1, 12}]];  (* A059317 *)

CROSSREFS

Cf. A229995.

Sequence in context: A338354 A014240 A014471 * A071028 A286987 A011635

Adjacent sequences:  A229999 A230000 A230001 * A230003 A230004 A230005

KEYWORD

tabf,sign,easy

AUTHOR

Clark Kimberling, Nov 07 2013

STATUS

approved

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Last modified December 3 02:46 EST 2021. Contains 349445 sequences. (Running on oeis4.)