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A086215 Number of (-1,0,1) n X n matrices M that are positive definite. 3

%I #19 Jan 11 2021 22:36:32

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

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

%C M need not be symmetric. For the number of different values of M + M' see A114601. - _Max Alekseyev_, Dec 13 2005

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PositiveDefiniteMatrix.html">Positive Definite Matrix</a>

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

%o (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_ */

%Y Cf. A114601, A085656.

%K nonn,hard,more

%O 1,2

%A _Eric W. Weisstein_, Jul 12 2003

%E a(4) from _Wouter Meeussen_, Sep 05 2003

%E a(5)-a(6) from _Max Alekseyev_, Dec 13 2005

%E a(7) from _Max Alekseyev_, Nov 09 2006

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