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A249303 Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments. 1

%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(n-1,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(n-1,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

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Last modified December 14 17:44 EST 2019. Contains 329979 sequences. (Running on oeis4.)