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 A383871 #14 Jul 23 2025 16:08:17
%S A383871 0,0,6,180,11720,3089250,5944080072,147348275209800,
%T A383871 38430603831264883632,90116197775746464859791750,
%U A383871 2118031078806486819496589635743440,966490887282837500134221233339527160717340,17165261053166610940029331024343115375665769316911576,6444206974822296283920298148689544172139277283018112679406098010
%N A383871 Number of labeled 3-nilpotent semigroups of order n.
%C A383871 A semigroup S is nilpotent if there exists a natural number r such that the set S^r of all products of r elements of S has size 1.
%C A383871 If r is the smallest such number, then S is said to have nilpotency degree r.
%C A383871 This sequence counts semigroups S that have an element e such that for all x,y,z in S x*y*z = e.
%C A383871 In 1976 Kleitman, Rothschild and Spencer gave an argument asserting that the proportion of 3-nilpotent semigroups, amongst all semigroups of order n, is asymptotically 1. Later opinion regards their argument as incomplete, and no satisfactory proof has been found.
%D A383871 H. Jürgensen, F. Migliorini, and J. Szép, Semigroups. Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1991.
%H A383871 Andreas Distler and James D. Mitchell, <a href="https://doi.org/10.48550/arXiv.1201.3529">The number of nilpotent semigroups of degree 3</a>, arXiv:1201.3529 [math.CO], 2012.
%H A383871 Igor Dolinka, D. G. FitzGerald, and James D. Mitchell, <a href="https://doi.org/10.48550/arXiv.2411.00466">Semirigidity and the enumeration of nilpotent semigroups of index three</a>, arXiv:2411.00466 [math.CO], 2024.
%H A383871 Pierre A. Grillet, <a href="https://doi.org/10.1080/00927872.2013.790036">Counting Semigroups</a>, Communications in Algebra, 43(2), 574-596, (2014).
%H A383871 D. J. Kleitman, B. R. Rothschild, and J. H. Spencer, <a href="https://doi.org/10.2307/2041879">The number of semigroups of order n</a>, Proc. Amer. Math. Soc. 55 (1976), 227-232.
%H A383871 <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Se#semigroups">Index entries for sequences related to semigroups</a>
%F A383871 a(n) = Sum_{2 <= m <= b(n)} binomial(n,m) * m * Sum_{0 <= i <= m-1} (-1)^i * binomial(m-1,i) * (m-i)^((n-m)^2), where b(n) = floor(n + 1/2 - sqrt(n-3/4)).
%Y A383871 Cf. A023814, A023815.
%Y A383871 Cf. A383885, A383886.
%K A383871 nonn
%O A383871 1,3
%A A383871 _Elijah Beregovsky_, May 13 2025