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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.

%I A350710 #46 May 22 2022 14:19:26
%S A350710 1,-1,1,-3,-2,1,-16,-16,-2,1,0,100,-10,-10,1,-1296,0,324,-24,-13,1,0,
%T A350710 0,4116,392,-175,-14,1,0,-131072,16384,12288,-512,-352,-12,1,0,0,
%U A350710 -708588,0,44469,2592,-459,-24,1,0,0,16000000,800000,-760000,-12000,11000,-100,-45,1
%N 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.
%e A350710 Triangle begins:
%e A350710 n=0:     1;
%e A350710 n=1:    -1,   1;
%e A350710 n=2:    -3,  -2,    1;
%e A350710 n=3:   -16, -16,   -2,   1;
%e A350710 n=4:     0, 100,  -10, -10,    1;
%e A350710 n=5: -1296,   0,  324, -24,  -13,   1;
%e A350710 n=6:     0,   0, 4116, 392, -175, -14, 1;
%e A350710 For example, the characteristic polynomial associated to M(7) is
%e A350710   q^7 - 12*q^6 - 352*q^5 - 512*q^4 + 12288*q^3 + 16384*q^2 - 131072*q + 0;
%e A350710 so the seventh row of the triangle is
%e A350710   0, -131072, 16384, 12288, -512, -352, -12, 1.
%p A350710 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(LinearAlgebra[
%p A350710     CharacteristicPolynomial](Matrix(n, (i, j)-> irem(i*j, n+1)), x)):
%p A350710 seq(T(n), n=0..10);  # _Alois P. Heinz_, Mar 27 2022
%t A350710 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}]
%o A350710 (PARI) row(n) = Vecrev(charpoly(matrix(n,n,i,j,i*j%(n+1)))); \\ _Kevin Ryde_, Mar 27 2022
%Y A350710 Cf. A352620 (matrices M).
%K A350710 sign,tabl
%O A350710 0,4
%A A350710 _Luca Onnis_, Mar 27 2022