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A360743 Number of idempotent binary relation matrices E on [n] such that E contains an identity matrix of order n-1 and (E - I_n)^2 = 0.

%I A360743 #30 Feb 27 2023 17:23:52
%S A360743 1,2,9,52,435,5046,81501,1823144,56572263,2435930410,145888123953,
%T A360743 12173595399516,1418664206897691,231298954644947294,
%U A360743 52860840028599821445,16957903154151836822608,7647128139328190245443279,4852236755345544324027858258
%N A360743 Number of idempotent binary relation matrices E on [n] such that E contains an identity matrix of order n-1 and (E - I_n)^2 = 0.
%C A360743 A Boolean relation matrix R is said to be convergent in its powers if in the sequence {R,R^2,R^3, ...} there is an m such that R^m = R^(m+1).
%C A360743 An idempotent Boolean relation matrix E is said to have a proper power primitive iff there is a convergent relation R with limit matrix E where R is not equal to E.
%C A360743 If an idempotent Boolean relation matrix E contains an identity matrix of order  n-1 and (E-I_n)^2 = 0 then E has no proper power primitive.  The converse is not true for n>=4. Consider {{1,0,1,0}, {0,1,0,1}, {0,0,0,0}, {0,0,0,0}}.  The converse is erroneously stated and proved in Rosenblatt, Theorem 4.
%H A360743 Alois P. Heinz, <a href="/A360743/b360743.txt">Table of n, a(n) for n = 0..113</a>
%H A360743 David Rosenblatt, <a href="http://dx.doi.org/10.6028/jres.067B.020">On the graphs of finite Boolean relation matrices</a>, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.
%F A360743 a(n) = (n + 1)*A001831(n).
%F A360743 E.g.f.: x*A'(x) + A(x) where A(x) = Sum_{n>=0} x^n/n! exp((2^n-1)*x) is the e.g.f. for A001831.
%p A360743 a:= n-> (n+1)*add(binomial(n, k)*(2^k-1)^(n-k), k=0..n):
%p A360743 seq(a(n), n=0..18);  # _Alois P. Heinz_, Feb 18 2023
%t A360743 nn = 16; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[A[x] + x D[A[x], x], {x, 0, nn}], x]
%Y A360743 Cf. A001831, A121337.
%K A360743 nonn
%O A360743 0,2
%A A360743 _Geoffrey Critzer_, Feb 18 2023
%E A360743 Corrected by _Geoffrey Critzer_, Feb 24 2023