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

1,2

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.

Links

Table of n, a(n) for n=1..10.

Example

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

Prog

(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)

Crossrefs

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

Sequence in context: A303927 A288693 A195895 * A175176 A362968 A233336

Adjacent sequences: A127499 A127500 A127501 * A127503 A127504 A127505

Keyword

nonn,nice

Author

Max Alekseyev, Jan 16 2007

Status

approved