A127502 - OEIS

%I #4 Mar 30 2012 17:26:59

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

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

%C 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.

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

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

%o (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)

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

%K nonn,nice

%O 1,2

%A _Max Alekseyev_, Jan 16 2007