A350710 Triangle read by rows formed from the coefficients in ascending order of the characteristic polynomial of the n X n matrix M(n) with entries M(n)[i,j] = i*j mod n+1.

1, -1, 1, -3, -2, 1, -16, -16, -2, 1, 0, 100, -10, -10, 1, -1296, 0, 324, -24, -13, 1, 0, 0, 4116, 392, -175, -14, 1, 0, -131072, 16384, 12288, -512, -352, -12, 1, 0, 0, -708588, 0, 44469, 2592, -459, -24, 1, 0, 0, 16000000, 800000, -760000, -12000, 11000, -100, -45, 1

Offset: 0

Author

Luca Onnis, Mar 27 2022

Keywords

• sign
• tabl

Examples

			Triangle begins:
n=0:     1;
n=1:    -1,   1;
n=2:    -3,  -2,    1;
n=3:   -16, -16,   -2,   1;
n=4:     0, 100,  -10, -10,    1;
n=5: -1296,   0,  324, -24,  -13,   1;
n=6:     0,   0, 4116, 392, -175, -14, 1;
For example, the characteristic polynomial associated to M(7) is
  q^7 - 12*q^6 - 352*q^5 - 512*q^4 + 12288*q^3 + 16384*q^2 - 131072*q + 0;
so the seventh row of the triangle is
  0, -131072, 16384, 12288, -512, -352, -12, 1.
		

Crossrefs

Cf. A352620 (matrices M).

Programs

Maple

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(LinearAlgebra[
    CharacteristicPolynomial](Matrix(n, (i, j)-> irem(i*j, n+1)), x)):
seq(T(n), n=0..10);  # Alois P. Heinz, Mar 27 2022

Mathematica

Table[(-1)^(p + 1)*CoefficientList[CharacteristicPolynomial[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], x], x], {p, 2, 20}]

PARI

row(n) = Vecrev(charpoly(matrix(n,n,i,j,i*j%(n+1)))); \\ Kevin Ryde, Mar 27 2022