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Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(x^2-x+1)).
2

%I #6 Nov 06 2019 19:14:51

%S -2,1,-1,4,-1,1,3,-6,1,2,-4,-6,8,-1,1,-10,10,10,-10,1,-1,-6,30,-20,

%T -15,12,-1,-2,7,21,-70,35,21,-14,1,-1,16,-28,-56,140,-56,-28,16,-1,1,

%U 9,-72,84,126,-252,84,36,-18,1,2,-10,-45,240,-210,-252,420,-120

%N Irregular triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of (x-2)/(x^2-x+1)).

%C Conjecture: The numbers n for which the n-th polynomial is irreducible are given by A069353.

%e First eight rows:

%e -2, 1;

%e -1, 4, -1;

%e 1, 3, -6, 1;

%e 2, -4, -6, 8, -1;

%e 1, -10, 10, 10, -10, 1;

%e -1, -6, 30, -20, -15, 12, -1;

%e -2, 7, 21, -70, 35, 21, -14, 1;

%e -1, 16, -28, -56, 140, -56, -28, 16, -1;

%e First eight polynomials:

%e -2 + x

%e -1 + 4 x - x^2

%e 1 + 3 x - 6 x^2 + x^3

%e 2 - 4 x - 6 x^2 + 8 x^3 - x^4

%e (1 + x) (1 - 11 x + 21 x^2 - 11 x^3 + x^4)

%e -1 - 6 x + 30 x^2 - 20 x^3 - 15 x^4 + 12 x^5 - x^6

%e (-2 + x) (1 - 3 x - 12 x^2 + 29 x^3 - 3 x^4 - 12 x^5 + x^6)

%e -1 + 16 x - 28 x^2 - 56 x^3 + 140 x^4 - 56 x^5 - 28 x^6 + 16 x^7 - x^8

%t g[x_, n_] := Numerator[ Factor[D[(x - 2)/(x^2 - x + 1), {x, n}]]]

%t Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A327315 polynomials *)

%t h[n_] := CoefficientList[g[x, n]/n!, x];

%t Table[h[n], {n, 0, 10}] (* A327315 sequence *)

%t Column[%] (* A327315 array *)

%Y Cf. A069353, A328646.

%K tabf,sign

%O 0,1

%A _Clark Kimberling_, Nov 01 2019