A292933 -id:A292933 - cp's OEIS frontend

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

Jean-Luc Marichal, Sep 27 2017

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

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.

Cf. A292933, A292934.

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

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

my(x='x + O('x^30)); Vec(serlaplace(1/(x+3-2*exp(x)))) \\ Michel Marcus, May 21 2019

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

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

A292934 E.g.f.: x^2/(x+3-2*exp(x)).

Original entry on oeis.org

0, 0, 2, 6, 48, 400, 4140, 49644, 681296, 10515600, 180349380, 3402380740, 70023004920, 1561206957336, 37485640585484, 964345394431020, 26462386594676640, 771532717446066208, 23817889096309943892, 776127882633846005268

Offset: 0

Jean-Luc Marichal, Sep 27 2017

Number of associative and quasitrivial binary operations on {1,...,n} that have both neutral and annihilator elements.

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

Cf. A292932, A292933.

a(n) = n*(n-1)*A292932(n-2).

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

A308352 Number of k-ary quasitrivial semigroups that have no neutral element on an n-element set.

Original entry on oeis.org

0, 2, 8, 58, 492, 5074, 60888, 835482, 12895796, 221169970, 4172486496, 85872215290, 1914575169756, 45970251182418, 1182618181384424, 32451961380002458, 946163712877067460, 29208900504551394610, 951798961321369842864, 32647628386008050898810

Offset: 1

J. Devillet, May 21 2019

Number of k-ary associative and quasitrivial operations that have no neutral element on an n-element set.

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).

Cf. A292932, A292933.

nmax = 20; Rest[CoefficientList[Series[(1 - x)/(3 - 2*E^x + x), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jun 05 2019 *)

seq(n)={Vec(-1+serlaplace((1-x)/(x+3-2*exp(x))) + O(x*x^n), -n)} \\ Andrew Howroyd, Aug 19 2019

a(n) = A292932(n) - n*A292932(n-1) = A292932(n) - A292933(n) for n >= 1.
a(n) ~ n! * (4-r) / ((r-1) * (r-3)^(n+1)), where r = -LambertW(-1, -2*exp(-3)). - Vaclav Kotesovec, Jun 05 2019
E.g.f.: (1 - x)/(x + 3 - 2*exp(x)). - Andrew Howroyd, Aug 19 2019

A308351 For n >= 2, a(n) = nu(n-1) + n(n-1)*u(n-2), where u = A292932; a(1)=1.

Original entry on oeis.org

1, 4, 18, 128, 1090, 11232, 134806, 1849696, 28550538, 489656720, 9237631150, 190115847792, 4238752713442, 101775333547552, 2618244556598310, 71846664091504064, 2094748778352174202, 64666725024407102064, 2107224874854168508126, 72279858915240296971600

Offset: 1

J. Devillet, May 21 2019

Robert Israel, Table of n, a(n) for n = 1..412
M. Couceiro, J. Devillet, All quasitrivial n-ary semigroups are reducible to semigroups, arXiv:1904.05968 [math.RA], 2019.

Cf. A292932, A292933.

E:= x*(1 + x)/(3 - 2*exp(x) + x):
S:= series(E,x,51):
seq(coeff(S,x,n)*n!,n=1..50); # Robert Israel, Nov 26 2020

nmax = 20; Rest[CoefficientList[Series[x*(1 + x)/(3 - 2*E^x + x), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jun 05 2019 *)

a(n) = n*A292932(n-1) + n*(n-1)*A292932(n-2) = A292933(n) + n*A292933(n-1) for n >= 2.
E.g.f.: x*(1 + x)/(3 - 2*exp(x) + x). - Vaclav Kotesovec, Jun 05 2019
a(n) ~ n! * (r-2) / ((r-1) * (r-3)^n), where r = -LambertW(-1, -2*exp(-3)). - Vaclav Kotesovec, Jun 05 2019

A308362 Number of (2k+1)-ary quasitrivial semigroups on an n-element set.

Original entry on oeis.org

1, 5, 23, 162, 1382, 14236, 170872, 2344530, 36188534, 620652000, 11708927276, 240976560622, 5372724404530, 129002764437228, 3318690040767224, 91067432174168202, 2655146132506208558, 81966680980803524728, 2670959894858615348356, 91616517379045122841830

Offset: 1

J. Devillet, May 22 2019

Number of (2k+1)-ary associative and quasitrivial operations on an n-element set.

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).

Cf. A308352, A308354, A292932, A292933.

nmax = 20; Rest[CoefficientList[Series[(2 + x^2)/(6 - 4*E^x + 2*x), {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, Jun 05 2019 *)

a(n) = A308352(n) + A292933(n) + A308354(n) for n >= 1.
a(n) = A292932(n) + binomial(n,2)*A292932(n-2) for n >= 2.
E.g.f.: (2 + x^2)/(6 - 4*exp(x) + 2*x). - Vaclav Kotesovec, Jun 05 2019
a(n) ~ n! * (r^2 - 6*r + 11) / (2*(r-1) * (r-3)^(n+1)), where r = -LambertW(-1, -2*exp(-3)). - Vaclav Kotesovec, Jun 05 2019

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A292934 E.g.f.: x^2/(x+3-2*exp(x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A308352 Number of k-ary quasitrivial semigroups that have no neutral element on an n-element set.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A308351 For n >= 2, a(n) = n*u(n-1) + n*(n-1)*u(n-2), where u = A292932; a(1)=1.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A308362 Number of (2k+1)-ary quasitrivial semigroups on an n-element set.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A308351 For n >= 2, a(n) = nu(n-1) + n(n-1)*u(n-2), where u = A292932; a(1)=1.