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%I #7 Jan 12 2024 16:06:58
%S 1,333,58335,2875405
%N Number of distinct characteristic polynomials for 4 X 4 matrices with entries from {0, 1, ..., n}.
%t mat[n_Integer?Positive] := mat[n] = Array[m, {n, n}];
%t flatMat[n_Integer?Positive] := flatMat[n] = Flatten[mat[n]];
%t charPolyMat[n_Integer?Positive] := charPolyMat[n] = FullSimplify[CoefficientList[Expand[CharacteristicPolynomial[mat[n], x]], x]];
%t a[d_Integer?Positive, 0] = 1; a[d_Integer?Positive, n_Integer?Positive] := a[d, n] = Length[DeleteDuplicates[Flatten[Table[Evaluate[charPolyMat[d]], ##] & @@ Table[{flatMat[d][[i]], 0, n}, {i, 1, d^2}], d^2 - 1]]];
%t Table[a[4, n], {n, 0, 2}]
%Y Cf. A366448 (2 X 2 matrices), A366551 (3 X 3 matrices).
%Y Cf. A272659.
%K nonn,more,hard
%O 0,2
%A _Robert P. P. McKone_, Dec 07 2023
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