cp's OEIS Frontend

A086193 Number of n X n matrices with entries in {0,1} with no zero row, no zero column and with zero main diagonal.

%I A086193 #40 Feb 16 2025 08:32:50
%S A086193 1,0,1,18,1699,592260,754179301,3562635108438,63770601591579079,
%T A086193 4405870283636411477640,1190873924687350003735546441,
%U A086193 1270602397076493907445608866890778,5381240610642043789096251476993474339179
%N A086193 Number of n X n matrices with entries in {0,1} with no zero row, no zero column and with zero main diagonal.
%C A086193 Also the number of simple labeled digraphs on n nodes for which every vertex has indegree at least one and outdegree at least one.
%C A086193 Also the number of edge covers on the n-crown graph. - _Eric W. Weisstein_, May 19 2017
%H A086193 Andrew Howroyd, <a href="/A086193/b086193.txt">Table of n, a(n) for n = 0..50</a>
%H A086193 R. W. Robinson. <a href="http://cobweb.cs.uga.edu/~rwr/publications/components.ps">Counting digraphs with restrictions on the strong components</a>, (1996). W(n).
%H A086193 R. W. Robinson, <a href="/A086193/a086193.pdf">Counting digraphs with restrictions on the strong components</a>, 1996 [Local copy, with permission]
%H A086193 Martin Svatoš, Peter Jung, Jan Tóth, Yuyi Wang, and Ondřej Kuželka, <a href="https://arxiv.org/abs/2302.04606">On Discovering Interesting Combinatorial Integer Sequences</a>, arXiv:2302.04606 [cs.LO], 2023, p. 17.
%H A086193 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CrownGraph.html">Crown Graph</a>
%H A086193 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EdgeCover.html">Edge Cover</a>
%F A086193 a(n) = Sum_{r=0..n} (-1)^(n-r)*binomial(n, r)*(2^(r-1)-1)^r*(2^r-1)^(n-r). - _Vladeta Jovovic_, Aug 27 2003
%F A086193 a(n) = sum( f(n, r), r=0..n ) where f(n, r) = binomial(n, r) (-1)^r (1-2^(-n+r+1))^(n-r) (1-2^(-n+r))^r 2^((n-r)(n-1)). - _Brendan McKay_, Aug 27 2003
%F A086193 E.g.f.: Sum_{k>=0} (2^(n-1)-1)^n*exp((1-2^n)*x)*x^n/n!. - _Vladeta Jovovic_, Feb 23 2008
%F A086193 a(n) ~ 2^(n*(n-1)). - _Vaclav Kotesovec_, May 04 2015
%t A086193 Table[ it = (Partition[ #1, n ] &) /@ IntegerDigits[ Range[ 0, -1 + 2^n^2 ], 2, n^2 ]; Count[ it, (q_)?MatrixQ /; Tr[ q ] === 0 && (Times @@ (Plus @@@ q)) > 0 && (Times @@ (Plus @@@ Transpose[ q ]) > 0) ], {n, 1, 4} ] (* _Wouter Meeussen_, Aug 25 2003 *)
%t A086193 Table[Sum[(-1)^(n-r)*Binomial[n, r]*(2^(r-1)-1)^r*(2^r-1)^(n-r), {r,0,n}],{n,1,15}] (* _Vaclav Kotesovec_, May 04 2015 after _Vladeta Jovovic_ *)
%o A086193 (PARI) a(n)={sum(r=0, n, (-1)^(n-r)*binomial(n, r)*(2^(r-1)-1)^r*(2^r-1)^(n-r))} \\ _Andrew Howroyd_, Sep 09 2018
%Y A086193 Cf. A048291.
%K A086193 nonn
%O A086193 0,4
%A A086193 _W. Edwin Clark_, Aug 25 2003
%E A086193 More terms from _Brendan McKay_, Aug 27 2003
%E A086193 a(0)=1 prepended by _Andrew Howroyd_, Sep 09 2018