A346209 Number of n X n matrices over GF(3) with no eigenvalues in GF(3), i.e., neither 0 nor 1 nor 2 is an eigenvalue.

1, 0, 18, 3456, 7619508, 149200289280, 26394940582090344, 42062797470468915399168, 603463180651533072058654437264, 77927374189849689541269666899007713280, 90570450400853976077932766909301405665963077152

Offset: 0

Author

Geoffrey Critzer, Jul 10 2021

Keywords

• nonn

Comments

Equivalently, a(n) is the number of n X n matrices over GF(3) whose characteristic polynomial has no linear factors.

Links

• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A002820, A051680, A027376, A053290.

Programs

Mathematica

nn = 10; q = 3; \[Nu] = Table[1/n Sum[MoebiusMu[n/d] q^d, {d, Divisors[n]}], {n, 1, nn}];Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[Product[1/(1 - u^d/q^(r d)), {r, 1, \[Infinity]}]^\[Nu][[d]], {d, 2, nn}], {u, 0, nn}], u]

Formula

Sum_{n>=0} a(n)*x^n/A053290(n) = Product_{d>=2} (Product_{r>=1} 1/(1-x^d/3^(r*d)))^A027376(d).