A086215 Number of (-1,0,1) n X n matrices M that are positive definite.

1, 7, 311, 79505, 105311665, 642005451319, 15477341239385927

Offset: 1

Author

Eric W. Weisstein, Jul 12 2003

Keywords

• nonn
• hard
• more

Comments

M need not be symmetric. For the number of different values of M + M' see A114601. - Max Alekseyev, Dec 13 2005

Links

• Eric Weisstein's World of Mathematics, Positive Definite Matrix

Crossrefs

Cf. A114601, A085656.

Programs

Mathematica

Table[Count[Tuples[{-1, 0, 1}, {n, n}], ?PositiveDefiniteMatrixQ], {n, 3}] (* _Eric W. Weisstein, Jan 03 2021 *)

PARI

{ a(n) = M=matrix(n,n,i,j,2*(i==j)); r=0; b(1); r } { b(k) = local(z,t); if(k>n, z=t=0; for(i=1,n, for(j=1,i-1, if(M[ i,j ]==0,z++); if(abs(M[ i, j ])==1,t++); )); r+=3^z*2^t; return; ); forvec(x=vector(k-1,i,[ -1,1 ]), for(i=1,k-1,M[ k,i ]=M[ i,k ]=x[ i ]); if( matdet(vecextract(M,2^k-1, 2^k-1),1)>0, b(k+1) ) ) } /* Max Alekseyev */

Extensions

a(4) from Wouter Meeussen, Sep 05 2003

a(5)-a(6) from Max Alekseyev, Dec 13 2005

a(7) from Max Alekseyev, Nov 09 2006