A308352 Number of k-ary quasitrivial semigroups that have no neutral element on an n-element set.

0, 2, 8, 58, 492, 5074, 60888, 835482, 12895796, 221169970, 4172486496, 85872215290, 1914575169756, 45970251182418, 1182618181384424, 32451961380002458, 946163712877067460, 29208900504551394610, 951798961321369842864, 32647628386008050898810

Offset: 1

Author

J. Devillet, May 21 2019

Keywords

• nonn
• easy

Comments

Number of k-ary associative and quasitrivial operations that have no neutral element on an n-element set.

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..413
• M. Couceiro, J. Devillet All quasitrivial n-ary semigroups are reducible to semigroups, arXiv:1904.05968 [math.RA], 2019.
• Jimmy Devillet, Miguel Couceiro, Characterizations and enumerations of classes of quasitrivial n-ary semigroups, 98th Workshop on General Algebra (AAA98, Dresden, Germany 2019).

Crossrefs

Cf. A292932, A292933.

Programs

Mathematica

nmax = 20; Rest[CoefficientList[Series[(1 - x)/(3 - 2*E^x + x), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jun 05 2019 *)

PARI

seq(n)={Vec(-1+serlaplace((1-x)/(x+3-2*exp(x))) + O(x*x^n), -n)} \\ Andrew Howroyd, Aug 19 2019

Formula

a(n) = A292932(n) - n*A292932(n-1) = A292932(n) - A292933(n) for n >= 1.

a(n) ~ n! * (4-r) / ((r-1) * (r-3)^(n+1)), where r = -LambertW(-1, -2*exp(-3)). - Vaclav Kotesovec, Jun 05 2019

E.g.f.: (1 - x)/(x + 3 - 2*exp(x)). - Andrew Howroyd, Aug 19 2019