A118994 Number of real n X n symmetric (+1,-1) matrices with positive determinant.

1, 0, 16, 432, 8448, 282240, 81949952, 32715189248, 12792558313472, 9318420858593280

Offset: 1

Author

Giovanni Resta, May 08 2006

Keywords

• nonn
• hard
• more

Crossrefs

Cf. A086900, A118992, A118995, A118997, A119003.

Programs

Maple

F:= proc(n) local Q,q,X,x,t,A,ii,L,v;
  Q:= [[1,1],seq(seq([i,j],i=2..j),j=2..n)];
  q:= nops(Q);
  X:= [seq(x[q[1],q[2]],q=Q)];
  t:= 0:
  A:= Matrix(n,n,shape=symmetric,symbol=x);
  A[2..n,1]:= Vector(n-1,1);
  for ii from 0 to 2^q-1 do
    L:= map(s -> 2*s-1, convert(2^q+ii,base,2)[1..q]);
    v:= LinearAlgebra:-Determinant(subs(zip(`=`,X,L),A));
    if v > 0 then t:= t+1 fi
  od;
  2^(n-1)*t;
end proc:
seq(F(n),n=1..7); # Robert Israel, Apr 14 2016

Formula

a(n) = A118992(n) - A118997(n). For odd n, a(n) = A118997(n) = A118992(n)/2. - Max Alekseyev, Jun 12 2025

Extensions

a(8) from Robert Israel, Apr 17 2016

a(9)-a(10) from Max Alekseyev, Jun 17 2025