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A085656 Number of positive-definite real {0,1} n X n matrices. 9
1, 3, 27, 681, 43369, 6184475, 1688686483, 665444089745 (list; graph; refs; listen; history; text; internal format)
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 A085657. - Max Alekseyev, Dec 13 2005

LINKS

Table of n, a(n) for n=1..8.

Index entries for sequences related to binary matrices

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, [0, 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) ) ) } (Alekseyev)

CROSSREFS

Cf. A055165, which counts nonsingular {0, 1} matrices and A085506, which counts {-1, 0, 1} matrices with positive eigenvalues.

Cf. A085657, A085658, A086215, A038379 (positive semi-definite matrices), A080858, A083029.

Sequence in context: A185037 A185237 A099084 * A113100 A038379 A047656

Adjacent sequences:  A085653 A085654 A085655 * A085657 A085658 A085659

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane, Jul 12 2003

EXTENSIONS

More terms from Max Alekseyev, Dec 13 2005

STATUS

approved

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Last modified January 17 10:30 EST 2019. Contains 319218 sequences. (Running on oeis4.)