A326926 Triangular array read by rows: row n shows the coefficients of this polynomial of degree n: (1/n!)*(numerator of n-th derivative of 1/(1-x+x^2)).

1, 1, -2, 0, -3, 3, -1, 0, 6, -4, -1, 5, 0, -10, 5, 0, 6, -15, 0, 15, -6, 1, 0, -21, 35, 0, -21, 7, 1, -8, 0, 56, -70, 0, 28, -8, 0, -9, 36, 0, -126, 126, 0, -36, 9, -1, 0, 45, -120, 0, 252, -210, 0, 45, -10, -1, 11, 0, -165, 330, 0, -462, 330, 0, -55, 11, 0

Offset: 0

Author

Clark Kimberling, Oct 24 2019

Keywords

• tabl
• sign

Comments

It appears that each nonconstant polynomial is irreducible if and only if its degree is p-1 for some prime p other than 3.

Examples

			First eight rows:
   1;
   1,  -2;
   0,  -3,   3;
  -1,   0,   6,  -4;
  -1,   5,   0, -10,   5;
   0,   6, -15,   0,  15,  -6;
   1,   0, -21,  35,   0, -21,   7;
   1,  -8,   0,  56, -70,   0,  28,  -8;
First eight polynomials:
1
1 - 2*x
-3*x + 3*x^2 = 3 (-1 + x)*x
-1 + 6*x^2 - 4*x^3 = (-1 + 2*x) (1 + 2*x - 2*x^2)
-1 + 5*x - 10*x^3 + 5*x^4
6*x - 15*x^2 + 15*x^4 - 6*x^5 = -3*x*(-2 + x)*(-1 + x)*(1 + x)*(-1 + 2*x)
1 - 21*x^2 + 35*x^3 - 21*x^5 + 7*x^6
1 - 8*x + 56*x^3 - 70*x^4 + 28*x^6 - 8*x^7 = -(-1 + 2*x)*(-1 - 2*x + 2*x^2)*(-1 + 8*x - 6*x^2 - 4*x^3 + 2*x^4)
		

Crossrefs

Cf. A326933.

Programs

Mathematica

g[x_, n_] := Numerator[ Factor[D[1/(x^2 - x + 1), {x, n}]]];
Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
h[n_] := CoefficientList[g[x, n]/n!, x]
Table[h[n], {n, 0, 10}]  (* A326926 *)
Column[%]
Table[-1 + Length[FactorList[g[x, n]/n!]], {n, 0, 100}]  (* A326933 *)