A229995 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).

1, 1, 0, 1, 1, 0, 3, 0, 1, 1, 0, 5, 0, 5, 0, 1, 1, 0, 7, 0, 13, 0, 7, 0, 1, 1, 0, 9, 0, 25, 0, 25, 0, 9, 0, 1, 1, 0, 11, 0, 41, 0, 63, 0, 41, 0, 11, 0, 1, 1, 0, 13, 0, 61, 0, 129, 0, 129, 0, 61, 0, 13, 0, 1, 1, 0, 15, 0, 85, 0, 231, 0, 321, 0, 231, 0, 85, 0

Offset: 0

Author

Clark Kimberling, Nov 07 2013

Keywords

• nonn
• tabf
• easy

Comments

Deleting the 0's leaves A008288 (Delannoy numbers as a triangle). 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).

Examples

			First 4 rows:
  1
  1 0 1
  1 0 3 0 1
  1 0 5 0 5 0 1
		

Crossrefs

Cf. A230002, A008288.

Programs

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 *)