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A292932 Number of quasitrivial semigroups on an arbitrary n-element set. 7
1, 1, 4, 20, 138, 1182, 12166, 146050, 2003882, 30930734, 530477310, 10007736906, 205965058162, 4592120925862, 110259944144486, 2836517343551714, 77836238876829882, 2269379773783175454, 70057736432648552782, 2282895953541692345722 (list; graph; refs; listen; history; text; internal format)
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 A098541 A004204

Adjacent sequences:  A292929 A292930 A292931 * A292933 A292934 A292935

KEYWORD

nonn,easy

AUTHOR

Jean-Luc Marichal, Sep 27 2017

STATUS

approved

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Last modified November 18 01:41 EST 2019. Contains 329242 sequences. (Running on oeis4.)