A086699 Number of n X n matrices over GF(2) with rank n-1.

1, 9, 294, 37800, 19373760, 39687459840, 325139829719040, 10654345790226432000, 1396491759480328106803200, 732164571206732295657278668800, 1535460761275478347250381697633484800, 12880379193826999985837000446453418557440000

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003

Keywords

• nonn

Comments

a(n)/2^(n^2) is the probability that a random linear operator T on an n dimensional vector space over the field with two elements is such that the dimension of the range of T equals n-1. This probability is Product{j>=2} 1 - 1/2^j which is 2 times the probability that the dimension of the range of T equals n. Cf. A048651. - Geoffrey Critzer, Jun 28 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..58

Crossrefs

Cf. A002884, A060867.

Programs

Mathematica

Table[Product[(q^n - q^i)^2/(q^(n - 1) - q^i), {i, 0, n - 2}] /.  q -> 2, {n, 0, 15}] (* Geoffrey Critzer, Jun 28 2017 *)

PARI

a(n) = prod(j=0, n-2, (2^n - 2^j)^2 / (2^(n-1)- 2^j)); \\ Michel Marcus, Jun 28 2017

Formula

for n>=2 : a(n) = product j=0...n-2 (2^n - 2^j)^2 / (2^(n-1)- 2^j).

Extensions

More terms from David Wasserman, Mar 28 2005