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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)).
3
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
0,3
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.
EXAMPLE
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)
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 *)
CROSSREFS
Cf. A326933.
Sequence in context: A094067 A094112 A333303 * A347564 A105569 A068455
KEYWORD
tabl,sign
AUTHOR
Clark Kimberling, Oct 24 2019
STATUS
approved