A230002 - OEIS

%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