A327315 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, 1, -1, 4, -1, 1, 3, -6, 1, 2, -4, -6, 8, -1, 1, -10, 10, 10, -10, 1, -1, -6, 30, -20, -15, 12, -1, -2, 7, 21, -70, 35, 21, -14, 1, -1, 16, -28, -56, 140, -56, -28, 16, -1, 1, 9, -72, 84, 126, -252, 84, 36, -18, 1, 2, -10, -45, 240, -210, -252, 420, -120

Offset: 0

Author

Clark Kimberling, Nov 01 2019

Keywords

• tabf
• sign

Comments

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

Examples

			First eight rows:
  -2,   1;
  -1,   4,  -1;
   1,   3,  -6,   1;
   2,  -4,  -6,   8,  -1;
   1, -10,  10,  10, -10,   1;
  -1,  -6,  30, -20, -15,  12,  -1;
  -2,   7,  21, -70,  35,  21, -14,  1;
  -1,  16, -28, -56, 140, -56, -28, 16, -1;
First eight polynomials:
  -2 + x
  -1 + 4 x - x^2
   1 + 3 x - 6 x^2 + x^3
   2 - 4 x - 6 x^2 + 8 x^3 - x^4
   (1 + x) (1 - 11 x + 21 x^2 - 11 x^3 + x^4)
  -1 - 6 x + 30 x^2 - 20 x^3 - 15 x^4 + 12 x^5 - x^6
   (-2 + x) (1 - 3 x - 12 x^2 + 29 x^3 - 3 x^4 - 12 x^5 + x^6)
  -1 + 16 x - 28 x^2 - 56 x^3 + 140 x^4 - 56 x^5 - 28 x^6 + 16 x^7 - x^8
		

Crossrefs

Cf. A069353, A328646.

Programs

Mathematica

g[x_, n_] := Numerator[ Factor[D[(x - 2)/(x^2 - x + 1), {x, n}]]]
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* A327315 polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x];
Table[h[n], {n, 0, 10}]  (* A327315 sequence *)
Column[%]   (* A327315 array *)