A346164 Number of n X n matrices over GF(2) whose characteristic polynomial is squarefree.

1, 2, 8, 160, 22272, 9744384, 20309999616, 165823024988160, 5334245506774204416, 699753231745207240753152, 366124801432291852761377538048, 769585907704777340287352115528990720, 6438115769123814066544745845515649290338304, 216154104085428332447218371078526172108859761491968

Offset: 0

Author

Geoffrey Critzer, Jul 08 2021

Keywords

• nonn

Links

• Jason Fulman Cycle Indices for the Finite Classical Groups, arXiv:math/9712239 [math.GR], 1997.
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A001037, A002884, A346082.

Programs

Mathematica

nn = 13; A001037 = Table[1/n Sum[MoebiusMu[n/d] 2^d, {d, Divisors[n]}], {n, 1, nn}]; Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[
  Series[Product[(1 + x^i/(2^i - 1))^A001037[[i]], {i, 1, nn}], {x, 0, nn}], x]

Formula

Sum_{n>=0} a(n)*x^n/A002884(n) = Product_{n>=1} (1 + x^n/(2^n-1))^A001037(n).

Lim_{n->infinity} a(n)/A002884(n) = 1. - Geoffrey Critzer, Oct 21 2021