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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. 3
1, 2, 9, 52, 435, 5046, 81501, 1823144, 56572263, 2435930410, 145888123953, 12173595399516, 1418664206897691, 231298954644947294, 52860840028599821445, 16957903154151836822608, 7647128139328190245443279, 4852236755345544324027858258 (list; graph; refs; listen; history; text; internal format)
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).
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.
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.
David Rosenblatt, On the graphs of finite Boolean relation matrices, Journal of Research, National Bureau of Standards, Vol 67B No. 4 Oct-Dec 1963.
a(n) = (n + 1)*A001831(n).
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.
a:= n-> (n+1)*add(binomial(n, k)*(2^k-1)^(n-k), k=0..n):
seq(a(n), n=0..18); # Alois P. Heinz, Feb 18 2023
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]
Sequence in context: A161631 A121678 A124347 * A360193 A347774 A360718
Geoffrey Critzer, Feb 18 2023
Corrected by Geoffrey Critzer, Feb 24 2023

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Last modified May 28 11:56 EDT 2024. Contains 372913 sequences. (Running on oeis4.)