A346165 Number of n X n invertible matrices whose characteristic polynomial is squarefree.

1, 1, 2, 104, 9792, 4887552, 10456694784, 80831009783808, 2695921347430711296, 347083584759711311855616, 184330749741189300682890412032, 383205061911277693825526401937178624, 3224343525101169010615339144085384529444864, 107976295438859678148286176040509108456782680817664

Offset

0,3

Links

Michael De Vlieger, Table of n, a(n) for n = 0..57

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.

Formula

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

Lim_{n->infinity} a(n)/A002884(n) = 1/2. - Geoffrey Critzer, Oct 21 2021

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

Crossrefs

Cf. A001037, A002884, A346082, A346164.

Sequence in context: A281053 A101530 A199945 * A265655 A001184 A370963

Adjacent sequences: A346162 A346163 A346164 * A346166 A346167 A346168

Keyword

nonn

Author

Geoffrey Critzer, Jul 08 2021

Status

approved