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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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..66.

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: A180013 A094067 A094112 * A105569 A068455 A329098

Adjacent sequences:  A326923 A326924 A326925 * A326927 A326928 A326929

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Oct 24 2019

STATUS

approved

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Last modified February 20 16:42 EST 2020. Contains 332080 sequences. (Running on oeis4.)