A308354 Number of (2k+1)-ary quasitrivial semigroups that have two neutral elements on an n-element set.

0, 1, 3, 24, 200, 2070, 24822, 340648, 5257800, 90174690, 1701190370, 35011502460, 780603478668, 18742820292742, 482172697215510, 13231193297338320, 385766358723033104, 11908944548154971946, 388063941316923002634, 13310969922203225028580

Offset: 1

Author

J. Devillet, May 21 2019

Keywords

• nonn
• easy

Comments

Number of (2k+1)-ary associative and quasitrivial operations that have two neutral elements 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.

Programs

Mathematica

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 *)

Formula

a(n) = binomial(n,2)*A292932(n-2) for n >= 2.

E.g.f.: x^2/(3 - 2*exp(x) + x)/2. - Vaclav Kotesovec, Jun 05 2019

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