0,3

Also the number of n X n real (0,1)-matrices with all eigenvalues positive.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 19, Eq. (1.6.1).

R. W. Robinson, Counting labeled acyclic digraphs, pp. 239-273 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.

T. D. Noe, Table of n, a(n) for n=0..40

B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices, J. Integer Sequences, 7 (2004), #04.3.3.

B. D. McKay, F. E. Oggier, G. F. Royle, N. J. A. Sloane, I. M. Wanless and H. S. Wilf, Acyclic digraphs and eigenvalues of (0,1)-matrices

Huantian Cao, AutoGF: An Automated System to Calculate Coefficients of Generating Functions.

Eric Weisstein's World of Mathematics, Positive Eigenvalued Matrix

Eric Weisstein's World of Mathematics, (0,1)-Matrix

Eric Weisstein's World of Mathematics, Acyclic Digraph

Index entries for sequences related to binary matrices

a(0) = 1; for n > 0, a(n) = Sum_{k=1..n} (-1)^(k+1)*C(n, k)*2^(k*(n-k))*a(n-k).

1 = Sum_{n>=0} a(n)*exp(-2^n*x)*x^n/n!. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 05 2005

a(n) = Sum_{k=1..n} (-1)^(n-k)*A046860(n,k) = Sum_{k=1..n} (-1)^(n-k)*k!*A058843(n,k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jun 20 2008

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

(PARI) a(n)=if(n<1, n==0, sum(k=1, n, -(-1)^k*C(n, k)*2^(k*(n-k))*a(n-k)))

Cf. A003087 (unlabeled DAGs), A086510.

Cf. A055165, which counts nonsingular {0,1} matrices, and A085656, which counts positive definite {0,1} matrices.

Sequence in context: A074708 A009843 A136173 this_sequence A131310 A127231 A062411

Adjacent sequences: A003021 A003022 A003023 this_sequence A003025 A003026 A003027

nonn,easy,nice

njas