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
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.
Amya Luo, Pattern Avoidance in Nonnesting Permutations, Undergraduate Thesis, Dartmouth College (2024). See p. 16.
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
KEYWORD
nonn,easy
AUTHOR
Jean-Luc Marichal, Sep 27 2017
STATUS
approved