A132186 Number of idempotent n X n matrices over GF(2); also number of diagonalizable n X n matrices over GF(2).

1, 2, 8, 58, 802, 20834, 1051586, 102233986, 19614424834, 7355623374338, 5494866505497602, 8087844439442585602, 23834930674299549249538, 138978138716920276085366786, 1626809921636911219317749563394, 37757678575184051755732304668884994

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..81

Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.

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

Formula

a(n) = sum(k=0...n, 2^(k(n-k))*[n,k]_2), where [n,k]_2 is the Gaussian binomial for q=2. - Marc van Leeuwen, May 22 2013

a(n)/A002884(n) is the coefficient of x^n in (Sum_{n>=0} x^n/A002884(n))^2. - Geoffrey Critzer, Aug 04 2017

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      `if`(n=0, 1, T(n-1, k-1)+2^k*T(n-1, k)))
    end:
a:= n-> add(2^(k*(n-k))*T(n, k), k=0...n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 06 2017

Mathematica

nn = 10; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 2; G[z_] := Sum[z^k/g[k], {k, 0, nn}]; Table[g[n], {n, 0, nn}] CoefficientList[Series[G[z]^2, {z, 0, nn}], z] (* Geoffrey Critzer, Aug 04 2017 *)
a[n_] := Block[{m}, Length@ Select[ Range[2^(n^2)], (m = Partition[ IntegerDigits[ #-1, 2, n^2], n]; Mod[m.m, 2] == m) &]]; a /@ Range[4] (* Giovanni Resta, Apr 09 2017 *)

Crossrefs

Cf. A002884, A081080.

Sequence in context: A179534 A256034 A086907 * A191603 A162068 A162069

Adjacent sequences: A132183 A132184 A132185 * A132187 A132188 A132189

Keyword

nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 19 2003 and Vladeta Jovovic, Nov 04 2007

Extensions

This is the result of merging two independently submitted but identical sequences. Thanks to Geoffrey Critzer for suggesting this. - N. J. A. Sloane, Dec 26 2017

Status

approved