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%I #14 Sep 17 2019 12:56:01
%S 1,2,8,96,1536,86016,5505024,1321205760,338228674560,335522845163520,
%T 343575393447444480,1385295986380096143360,5674172360212873803202560,
%U 92239345887620476544860815360,1511249443022773887710999598858240,98654363640526679389774053813465907200
%N 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).
%C 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.
%F For even n = 2m, a(n) = 2^(m^2) * |GL(m, 2)| = 2^(m^2) * A002884(m).
%F For odd n = 2m+1, a(n) = 2^(m^2+2m) * |GL(m, 2)| = 2^(m^2+2m) * A002884(m).
%t a[n_] := With[{m = Quotient[n, 2]}, 2^(2 m^2 + 2 m Boole[OddQ[n]]) * QPochhammer[2^-m, 2, m]];
%t a /@ Range[1, 16] (* _Jean-François Alcover_, Sep 17 2019 *)
%o (GAP)
%o a:=function(n) local M;
%o M:=NullMat(n,n); for i in [1..n] do M[i][n+1-i]:=1; od;
%o return Size(Centralizer(GL(n, Integers mod 2), M * One(Integers mod 2)));
%o end; # _Andrew Howroyd_, Jul 13 2018
%o (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
%Y Cf. A002884, A087918.
%K nonn
%O 1,2
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 24 2003
%E a(8)-a(16) from _Andrew Howroyd_, Jul 13 2018
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