This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360718 #41 Feb 15 2024 02:43:58
%S A360718 1,2,9,52,459,5526,91161,2039024,62264215,2618031658,153147765333,
%T A360718 12544274587956,1443661355799075,233590364506712318,
%U A360718 53152637809972391281,17010099259539378971368,7660283773351147860024879,4856904906875123474086041426
%N A360718 Number of idempotent Boolean relation matrices on [n] that have no proper power primitive.
%C A360718 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 A360718 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 A360718 Let P = C_1 + C_2 + ... + C_k + S be a poset with rank(P) <= 1 (A001831) where each C_i is a weakly connected component of size 2 or more and S is a set of isolated points. Let A be a subset of [n] and let E = P - {(x, x): x in A}. Then E is an idempotent relation with no proper power primitive iff A satisfies exactly one of the following conditions:
%C A360718 i) A is a nonempty subset of domain(E) and A contains at most one point in domain(C_i) for 1 <= i <= k.
%C A360718 ii) A is a nonempty subset of image(E) and A contains at most one point in image(C_i) for 1 <= i <= k.
%C A360718 iii) A contains at most one point in S.
%C A360718 The first term in the e.g.f. below counts the number of such relations for which condition i) or ii) is satisfied. The second term in the e.g.f. counts the number of such relations for which condition iii) is satisfied. - _Geoffrey Critzer_, Feb 11 2024
%H A360718 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 A360718 E.g.f.: 2(exp(x * c'(x)/2) - 1) exp(c(x)) exp(x) + exp(c(x))*(x exp(x))' where c(x) is the e.g.f. for A002031.
%t A360718 nn = 17; A[x_] := Sum[x^n/n! Exp[(2^n - 1) x], {n, 0, nn}]; c[x_] := Log[A[x]] - x; Range[0, nn]! CoefficientList[Series[2 (Exp[x D[c[x], x]/2] - 1) Exp[c[x]] Exp[x] + Exp[c[x]] D[x Exp[x], x], {x, 0, nn}], x]
%Y A360718 Cf. A360743, A001831, A121337, A002031.
%K A360718 nonn
%O A360718 0,2
%A A360718 _Geoffrey Critzer_, Feb 24 2023