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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).

%I A229995 #11 Feb 24 2025 06:27:29
%S A229995 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,
%T A229995 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,
%U A229995 13,0,1,1,0,15,0,85,0,231,0,321,0,231,0,85,0
%N 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).
%C A229995 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).
%e A229995 First 4 rows:
%e A229995   1
%e A229995   1 0 1
%e A229995   1 0 3 0 1
%e A229995   1 0 5 0 5 0 1
%t A229995 p[n_, x_] := p[x] = Fibonacci[n, x]; Table[p[n, x], {n, 1, 10}]
%t A229995 f[n_, x_] := f[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1/x]]]
%t A229995 g[n_, x_] := g[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x - 1/x]]]
%t A229995 h[n_, x_] := h[n, x] = Expand[Numerator[Factor[p[n, x] /. x -> x + 1 + 1/x]]]
%t A229995 t1 = Flatten[Table[CoefficientList[f[n, x], x], {n, 1, 12}]];  (* A229995 *)
%t A229995 t2 = Flatten[Table[CoefficientList[g[n, x], x], {n, 1, 12}]];  (* A230002 *)
%t A229995 t3 = Flatten[Table[CoefficientList[h[n, x], x], {n, 1, 12}]];  (* A059317 *)
%Y A229995 Cf. A230002, A008288.
%K A229995 nonn,tabf,easy
%O A229995 0,7
%A A229995 _Clark Kimberling_, Nov 07 2013