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%I #12 Jul 21 2021 09:27:50
%S 1,0,18,3456,7619508,149200289280,26394940582090344,
%T 42062797470468915399168,603463180651533072058654437264,
%U 77927374189849689541269666899007713280,90570450400853976077932766909301405665963077152
%N 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.
%C Equivalently, a(n) is the number of n X n matrices over GF(3) whose characteristic polynomial has no linear factors.
%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F Sum_{n>=0} a(n)*x^n/A053290(n) = Product_{d>=2} (Product_{r>=1} 1/(1-x^d/3^(r*d)))^A027376(d).
%t 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]
%Y Cf. A002820, A051680, A027376, A053290.
%K nonn
%O 0,3
%A _Geoffrey Critzer_, Jul 10 2021
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