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A360743
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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.
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3
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1, 2, 9, 52, 435, 5046, 81501, 1823144, 56572263, 2435930410, 145888123953, 12173595399516, 1418664206897691, 231298954644947294, 52860840028599821445, 16957903154151836822608, 7647128139328190245443279, 4852236755345544324027858258
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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MAPLE
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a:= n-> (n+1)*add(binomial(n, k)*(2^k-1)^(n-k), k=0..n):
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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