A087540 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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