A127502 Number of n X n positive definite matrices with 1's on the main diagonal and -1's and 0's elsewhere.

1, 3, 19, 201, 3001, 55291, 1115003, 21837649, 373215601, 8282131891

Offset: 1

Author

Max Alekseyev, Jan 16 2007

Keywords

• nonn
• nice

Comments

A real matrix M is positive-definite if x M x' > 0 for all nonzero real vectors x. Equivalently, all eigenvalues of M + M' are positive.

M need not be symmetric. For the number of different values of M + M' see A084552.

Examples

			For n = 2 the three matrices are {{{1, 0}, {0, 1}}, {{1, 0}, {-1, 1}}, {{1, -1}, {0, 1}}}.
		

Crossrefs

Cf. A085656, A085657, A085658, A086215, A038379, A080858, A083029, A127503.

Programs

PARI

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