|
%I #12 Oct 31 2019 21:42:05
%S 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,
%T -21,7,1,-8,0,56,-70,0,28,-8,0,-9,36,0,-126,126,0,-36,9,-1,0,45,-120,
%U 0,252,-210,0,45,-10,-1,11,0,-165,330,0,-462,330,0,-55,11,0
%N 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)).
%C It appears that each nonconstant polynomial is irreducible if and only if its degree is p-1 for some prime p other than 3.
%e First eight rows:
%e 1;
%e 1, -2;
%e 0, -3, 3;
%e -1, 0, 6, -4;
%e -1, 5, 0, -10, 5;
%e 0, 6, -15, 0, 15, -6;
%e 1, 0, -21, 35, 0, -21, 7;
%e 1, -8, 0, 56, -70, 0, 28, -8;
%e First eight polynomials:
%e 1
%e 1 - 2*x
%e -3*x + 3*x^2 = 3 (-1 + x)*x
%e -1 + 6*x^2 - 4*x^3 = (-1 + 2*x) (1 + 2*x - 2*x^2)
%e -1 + 5*x - 10*x^3 + 5*x^4
%e 6*x - 15*x^2 + 15*x^4 - 6*x^5 = -3*x*(-2 + x)*(-1 + x)*(1 + x)*(-1 + 2*x)
%e 1 - 21*x^2 + 35*x^3 - 21*x^5 + 7*x^6
%e 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)
%t g[x_, n_] := Numerator[ Factor[D[1/(x^2 - x + 1), {x, n}]]];
%t Column[Expand[Table[g[x, n]/n!, {n, 0, 12}]]] (* polynomials *)
%t h[n_] := CoefficientList[g[x, n]/n!, x]
%t Table[h[n], {n, 0, 10}] (* A326926 *)
%t Column[%]
%t Table[-1 + Length[FactorList[g[x, n]/n!]], {n, 0, 100}] (* A326933 *)
%Y Cf. A326933.
%K tabl,sign
%O 0,3
%A _Clark Kimberling_, Oct 24 2019
|
