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%I #6 Nov 09 2013 03:42:05
%S 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,
%T 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,
%U -1,0,1,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0
%N 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).
%C 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.
%e First 5 rows: (1}, (-1,0,1), (1,0,-1,0,1), (-1,0,1,0,-1,0,1).
%e First 5 polynomials: 1, -1 + x^2, 1 - x^2 + x^4, -1 + x^2 - x^4 + x^6.
%t p[n_, x_] := p[x] = Fibonacci[n, x]; Table[p[n, x], {n, 1, 10}]
%t f[n_, x_] := f[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1/x]]]
%t g[n_, x_] := g[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x - 1/x]]]
%t h[n_, x_] := h[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1 + 1/x]]]
%t t1 = Flatten[Table[CoefficientList[f[n, x], x], {n, 1, 12}]]; (* A229995 *)
%t t2 = Flatten[Table[CoefficientList[g[n, x], x], {n, 1, 12}]]; (* A230002 *)
%t t3 = Flatten[Table[CoefficientList[h[n, x], x], {n, 1, 12}]]; (* A059317 *)
%Y Cf. A229995.
%K tabf,sign,easy
%O 0
%A _Clark Kimberling_, Nov 07 2013
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