A292932 Number of quasitrivial semigroups on an arbitrary n-element set.

1, 1, 4, 20, 138, 1182, 12166, 146050, 2003882, 30930734, 530477310, 10007736906, 205965058162, 4592120925862, 110259944144486, 2836517343551714, 77836238876829882, 2269379773783175454, 70057736432648552782, 2282895953541692345722

Offset

0,3

Comments

Number of associative and quasitrivial binary operations on {1,...,n}. Convention a(0)=1.

Links

G. C. Greubel, Table of n, a(n) for n = 0..410

M. Couceiro, J. Devillet, and J.-L. Marichal, Quasitrivial semigroups: characterizations and enumerations, arXiv:1709.09162 [math.RA], 2017.

Jimmy Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017.

Jimmy Devillet, Miguel Couceiro, Characterizations and enumerations of classes of quasitrivial n-ary semigroups, 98th Workshop on General Algebra (AAA98, Dresden, Germany 2019).

Jimmy Devillet, Jean-Luc Marichal, Bruno Teheux, Classifications of quasitrivial semigroups, arXiv:1811.11113 [math.RA], 2018.

Formula

E.g.f.: 1/(3 + x - 2*exp(x)).

Recurrence: a(0) = 1, a(n+1) = (n+1)*a(n) + 2*Sum_{k=0...n-1} binomial(n+1,k)*a(k).

Explicit form: a(n) = Sum_{i=0...n} Sum_{k=0...n-i} 2^i * (-1)^k * binomial(n,k) * S2(n-k,i) * (i+k)!, where S2(n,k) are the Stirling numbers of the second kind.

a(n) ~ n! / ((r-1) * (r-3)^(n+1)), where r = -LambertW(-1, -2*exp(-3)) = 3.5830738760366909976807989989303134394318270218566... - Vaclav Kotesovec, Sep 27 2017

Mathematica

With[{m=30}, CoefficientList[Series[1/(3+x-2*Exp[x]), {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 21 2019 *)

Prog

(PARI) my(x='x + O('x^30)); Vec(serlaplace(1/(x+3-2*exp(x)))) \\ Michel Marcus, May 21 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( 1/(3+x-2*Exp(x)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 21 2019
(Sage) m = 30; T = taylor(1/(3+x-2*exp(x)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 21 2019

Crossrefs

Cf. A292933, A292934.

Sequence in context: A129102 A129099 A244756 * A187116 A340903 A098541

Adjacent sequences: A292929 A292930 A292931 * A292933 A292934 A292935

Keyword

nonn,easy

Author

Jean-Luc Marichal, Sep 27 2017

Status

approved