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%I #20 Aug 19 2019 16:50:34
%S 0,2,8,58,492,5074,60888,835482,12895796,221169970,4172486496,
%T 85872215290,1914575169756,45970251182418,1182618181384424,
%U 32451961380002458,946163712877067460,29208900504551394610,951798961321369842864,32647628386008050898810
%N Number of k-ary quasitrivial semigroups that have no neutral element on an n-element set.
%C Number of k-ary associative and quasitrivial operations that have no neutral element on an n-element set.
%H Michael De Vlieger, <a href="/A308352/b308352.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) = A292932(n) - n*A292932(n-1) = A292932(n) - A292933(n) for n >= 1.
%F a(n) ~ n! * (4-r) / ((r-1) * (r-3)^(n+1)), where r = -LambertW(-1, -2*exp(-3)). - _Vaclav Kotesovec_, Jun 05 2019
%F E.g.f.: (1 - x)/(x + 3 - 2*exp(x)). - _Andrew Howroyd_, Aug 19 2019
%t nmax = 20; Rest[CoefficientList[Series[(1 - x)/(3 - 2*E^x + x), {x, 0, nmax}], x] * Range[0, nmax]!] (* _Vaclav Kotesovec_, Jun 05 2019 *)
%o (PARI) seq(n)={Vec(-1+serlaplace((1-x)/(x+3-2*exp(x))) + O(x*x^n), -n)} \\ _Andrew Howroyd_, Aug 19 2019
%Y Cf. A292932, A292933.
%K nonn,easy
%O 1,2
%A _J. Devillet_, May 21 2019
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