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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
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.
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
Sequence in context: A126429 A349267 A098272 * A052713 A136797 A255132
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