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Number of real {0,1} n X n matrices A such that M = A + A' has 2's on the main diagonal, 0's and 1's elsewhere and is positive semi-definite.
5

%I #20 Jun 23 2024 11:51:34

%S 1,3,27,729,52649,9058475,3383769523,2520512534065

%N Number of real {0,1} n X n matrices A such that M = A + A' has 2's on the main diagonal, 0's and 1's elsewhere and is positive semi-definite.

%C Necessarily A has all 1's on the main diagonal.

%C A real matrix M is positive semi-definite if its eigenvalues are >= 0.

%C For n <= 4, a(n) equals the upper bound 3^C(n,2).

%C For the number of different values of symmetric parts A + A', see A085658. - _Max Alekseyev_, Nov 11 2006

%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>

%F a(n) = Sum_{k=0..C(n,2)} 2^k * A083029(n,k).

%Y Cf. A055165, which counts nonsingular {0, 1} matrices, A003024, which counts {0, 1} matrices with positive eigenvalues, A085656 (positive definite matrices).

%Y Cf. A085657, A085658, A080858, A083029.

%K nonn,more,nice

%O 1,2

%A _N. J. A. Sloane_, Jul 13 2003

%E Definition corrected Nov 10 2006

%E a(6)-a(8) from _Max Alekseyev_, Nov 11 2006

%E Edited by _Max Alekseyev_, Jun 05 2024