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%I #26 Dec 14 2018 09:19:07
%S 1,2,13,261,15418,2566333
%N Number of n X n real (0,1)-matrices with all eigenvalues >= 0.
%H B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, <a href="https://arxiv.org/abs/math/0310423">Acyclic digraphs and eigenvalues of (0,1)-matrices</a>, arXiv:math/0310423 [math.CO], 2003.
%H B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sloane/sloane15.html">Acyclic digraphs and eigenvalues of (0,1)-matrices</a>, J. Integer Sequences, 7 (2004), #04.3.3.
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%e a(2)=13 because only 3 of the 16 possible matrices have eigenvalues < 0:
%e .
%e 0 1
%e 1 0
%e with eigenvalues {1,-1}
%e and
%e 1 1
%e 1 0
%e .
%e 0 1
%e 1 1
%e both with eigenvalues {1.61803..(Golden ratio),-0.61803...}
%t a[0] = 1; a[n_] := Module[{M, iter, cnt = 0}, M = Table[a[i, j], {i, 1, n}, {j, 1, n}]; iter = Thread[{Flatten[M], 0, 1}]; Do[If[AllTrue[Eigenvalues[ M], NonNegative], cnt++], Evaluate[Sequence @@ iter]]; cnt];
%t Do[Print[n, " ", a[n]], {n, 0, 5}] (* _Jean-François Alcover_, Dec 09 2018 *)
%Y Cf. A003024, A055165, A085656, A098148.
%K nonn,more
%O 0,2
%A Frederique Oggier (frederique.oggier(AT)epfl.ch) and _N. J. A. Sloane_, Sep 10 2003
%E a(5) from _Hugo Pfoertner_, Sep 26 2017
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