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A249303 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

%I A249303 #10 Feb 11 2020 02:04:24
%S A249303 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 A249303 -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 A249303 14,7,0,5,-30,60,-35,-21,21,1,-1,10,-30,20,50,-84,28,8
%N A249303 Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.
%C A249303 The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = 1 + (x - 1)/f(n-1,x), where f(0,x) = 1.
%C A249303 Every row sum is 1.  The first column is purely periodic with period (1,0,-1,-1,0,1).
%C A249303 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(n-1,x), where f(x,0) = 1, then p(n,x) is irreducible if and only if n is a (prime - 2).
%H A249303 Clark Kimberling, <a href="/A249303/b249303.txt">Rows 0..100, flattened</a>
%e A249303 f(0,x) = 1/1, so that p(0,x) = 1
%e A249303 f(1,x) = x/1, so that p(1,x) = x;
%e A249303 f(2,x) = (-1 + 2 x)/x, so that p(2,x) = -1 + 2 x.
%e A249303 First 6 rows of the triangle of coefficients:
%e A249303 ... 1
%e A249303 ... 0 ... 1
%e A249303 .. -1 ... 2
%e A249303 .. -1 ... 1 ... 1
%e A249303 ... 0 .. -2 ... 3
%e A249303 ... 1 .. -4 ... 3 ... 1
%t A249303 z = 20; f[n_, x_] := 1 + (x - 1)/f[n - 1, x]; f[0, x_] = 1;
%t A249303 t = Table[Factor[f[n, x]], {n, 0, z}]
%t A249303 u = Numerator[t]
%t A249303 TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]] (* A249303 array *)
%t A249303 v = Flatten[CoefficientList[u, x]] (* A249303 *)
%Y A249303 Cf. A128100, A229057, A229074.
%K A249303 tabf,sign,easy
%O A249303 0,5
%A A249303 _Clark Kimberling_, Oct 24 2014