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Number of real n X n symmetric (+1,-1) matrices with positive determinant.
3

%I #9 Apr 17 2016 09:35:14

%S 1,0,16,432,8448,282240,81949952,32715189248

%N Number of real n X n symmetric (+1,-1) matrices with positive determinant.

%p F:= proc(n) local Q,q,X,x,t,A,ii,L,v;

%p Q:= [[1,1],seq(seq([i,j],i=2..j),j=2..n)];

%p q:= nops(Q);

%p X:= [seq(x[q[1],q[2]],q=Q)];

%p t:= 0:

%p A:= Matrix(n,n,shape=symmetric,symbol=x);

%p A[2..n,1]:= Vector(n-1,1);

%p for ii from 0 to 2^q-1 do

%p L:= map(s -> 2*s-1, convert(2^q+ii,base,2)[1..q]);

%p v:= LinearAlgebra:-Determinant(subs(zip(`=`,X,L),A));

%p if v > 0 then t:= t+1 fi

%p od;

%p 2^(n-1)*t;

%p end proc:

%p seq(F(n),n=1..7); # _Robert Israel_, Apr 14 2016

%Y Cf. A086900, A118995, A118997.

%K nonn,hard,more

%O 1,3

%A _Giovanni Resta_, May 08 2006

%E a(8) from _Robert Israel_, Apr 17 2016