A346165 - OEIS

%I #15 Oct 22 2021 13:49:38

%S 1,1,2,104,9792,4887552,10456694784,80831009783808,

%T 2695921347430711296,347083584759711311855616,

%U 184330749741189300682890412032,383205061911277693825526401937178624,3224343525101169010615339144085384529444864,107976295438859678148286176040509108456782680817664

%N Number of n X n invertible matrices whose characteristic polynomial is squarefree.

%H Michael De Vlieger, <a href="/A346165/b346165.txt">Table of n, a(n) for n = 0..57</a>

%H Jason Fulman <a href="https://arxiv.org/abs/math/9712239">Cycle Indices for the Finite Classical Groups</a>, arXiv:math/9712239 [math.GR], 1997.

%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/A002884(n) = (1 + x) * Product_{n>=2} (1 + x^n/(2^n -1))^A001037(n)

%F Lim_{n->infinity} a(n)/A002884(n) = 1/2. - _Geoffrey Critzer_, Oct 21 2021

%t 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[

%t   Series[(1 + x) Product[(1 + x^i/(2^i - 1))^A001037[[i]], {i, 2,nn}], {x, 0, nn}], x]

%Y Cf. A001037, A002884, A346082, A346164.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Jul 08 2021