%I
%S 1,0,1,1,2,1,1,1,0,2,3,1,4,3,1,1,2,2,4,0,3,9,6,1,1,6,9,0,5,
%T 1,3,3,15,10,1,0,4,18,24,5,6,1,8,18,6,20,15,1,1,4,4,36,49,
%U 14,7,0,5,30,60,35,21,21,1,1,10,30,20,50,84,28,8
%N Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = 1 + (x  1)/f(n1,x), where f(0,x) = 1.
%C Every row sum is 1. The first column is purely periodic with period (1,0,1,1,0,1).
%C Conjecture: for n > 2, p(n,x) is irreducible if and only if n is a (prime  2). More generally, if c is arbitrary and f(n,x) = 1 + (x + c)/f(n1,x), where f(x,0) = 1, then p(n,x) is irreducible if and only if n is a (prime  2).
%H Clark Kimberling, <a href="/A249303/b249303.txt">Rows 0..100, flattened</a>
%e f(0,x) = 1/1, so that p(0,x) = 1
%e f(1,x) = x/1, so that p(1,x) = x;
%e f(2,x) = (1 + 2 x)/x, so that p(2,x) = 1 + 2 x.
%e First 6 rows of the triangle of coefficients:
%e ... 1
%e ... 0 ... 1
%e .. 1 ... 2
%e .. 1 ... 1 ... 1
%e ... 0 .. 2 ... 3
%e ... 1 .. 4 ... 3 ... 1
%t z = 20; f[n_, x_] := 1 + (x  1)/f[n  1, x]; f[0, x_] = 1;
%t t = Table[Factor[f[n, x]], {n, 0, z}]
%t u = Numerator[t]
%t TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]] (*A249303 array*)
%t v = Flatten[CoefficientList[u, x]] (*A249303*)
%Y Cf. A128100, A229057, A229074.
%K tabf,sign,easy
%O 0,5
%A _Clark Kimberling_, Oct 24 2014
