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
Links
- G. C. Greubel, Table of n, a(n) for n = 0..410
- Miguel Couceiro, Jimmy Devillet, and Jean-Luc 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 and 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, and Bruno Teheux, Classifications of quasitrivial semigroups, arXiv:1811.11113 [math.RA], 2018.
Programs
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Magma
m:=30; R
:=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 -
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 *) -
PARI
my(x='x + O('x^30)); Vec(serlaplace(1/(x+3-2*exp(x)))) \\ Michel Marcus, 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
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
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