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%I #22 Aug 19 2019 16:50:42
%S 0,1,3,24,200,2070,24822,340648,5257800,90174690,1701190370,
%T 35011502460,780603478668,18742820292742,482172697215510,
%U 13231193297338320,385766358723033104,11908944548154971946,388063941316923002634,13310969922203225028580
%N Number of (2k+1)-ary quasitrivial semigroups that have two neutral elements on an n-element set.
%C Number of (2k+1)-ary associative and quasitrivial operations that have two neutral elements on an n-element set.
%H Michael De Vlieger, <a href="/A308354/b308354.txt">Table of n, a(n) for n = 1..413</a>
%H M. Couceiro, J. Devillet <a href="https://arxiv.org/abs/1904.05968">All quasitrivial n-ary semigroups are reducible to semigroups</a>, arXiv:1904.05968 [math.RA], 2019.
%H Jimmy Devillet, Miguel Couceiro, <a href="http://orbilu.uni.lu/handle/10993/39720">Characterizations and enumerations of classes of quasitrivial n-ary semigroups</a>, 98th Workshop on General Algebra (AAA98, Dresden, Germany 2019).
%F a(n) = binomial(n,2)*A292932(n-2) for n >= 2.
%F E.g.f.: x^2/(3 - 2*exp(x) + x)/2. - _Vaclav Kotesovec_, Jun 05 2019
%F a(n) ~ n! / (2*(r-1) * (r-3)^(n-1)), where r = -LambertW(-1, -2*exp(-3)). - _Vaclav Kotesovec_, Jun 05 2019
%t nmax = 20; Rest[CoefficientList[Series[x^2/(3 - 2*E^x + x)/2, {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Jun 05 2019 *)
%Y Cf. A292932.
%K nonn,easy
%O 1,3
%A _J. Devillet_, May 21 2019
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