A087540 Let A(n) be the matrix in the group GL(n,2) such that for 1 <= i, j <= n: A[i,j] = 1 if i+j = n+1 A[i,j] = 0 if i+j != n+1. a(n) is the size of the centralizer of A(n) in GL(n,2).

1, 2, 8, 96, 1536, 86016, 5505024, 1321205760, 338228674560, 335522845163520, 343575393447444480, 1385295986380096143360, 5674172360212873803202560, 92239345887620476544860815360, 1511249443022773887710999598858240, 98654363640526679389774053813465907200

Offset

1,2

Comments

The formula was given by Derek Holt (mareg(AT)mimosa.csv.warwick.ac.uk) in this thread from sci.math: http://mathforum.org/discuss/sci.math/t/538859.

Links

Table of n, a(n) for n=1..16.

Formula

For even n = 2m, a(n) = 2^(m^2) * |GL(m, 2)| = 2^(m^2) * A002884(m).

For odd n = 2m+1, a(n) = 2^(m^2+2m) * |GL(m, 2)| = 2^(m^2+2m) * A002884(m).

Mathematica

a[n_] := With[{m = Quotient[n, 2]}, 2^(2 m^2 + 2 m Boole[OddQ[n]]) * QPochhammer[2^-m, 2, m]];
a /@ Range[1, 16] (* Jean-François Alcover, Sep 17 2019 *)

Prog

(GAP)
a:=function(n) local M;
M:=NullMat(n, n); for i in [1..n] do M[i][n+1-i]:=1; od;
return Size(Centralizer(GL(n, Integers mod 2), M * One(Integers mod 2)));
end; # Andrew Howroyd, Jul 13 2018
(PARI) a(n)={my(m=n\2); 2^(m*if(n%2, n+3, n)/2)*prod(i=2, m, 2^i-1)*2^binomial(m, 2)} \\ Andrew Howroyd, Jul 13 2018

Crossrefs

Cf. A002884, A087918.

Sequence in context: A126429 A349267 A098272 * A052713 A136797 A255132

Adjacent sequences: A087537 A087538 A087539 * A087541 A087542 A087543

Keyword

nonn

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 24 2003

Extensions

a(8)-a(16) from Andrew Howroyd, Jul 13 2018

Status

approved