J. Devillet - cp's OEIS frontend

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

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

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

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

A308354 Number of (2k+1)-ary quasitrivial semigroups that have two neutral elements on an n-element set.

Original entry on oeis.org

0, 1, 3, 24, 200, 2070, 24822, 340648, 5257800, 90174690, 1701190370, 35011502460, 780603478668, 18742820292742, 482172697215510, 13231193297338320, 385766358723033104, 11908944548154971946, 388063941316923002634, 13310969922203225028580

Offset: 1

J. Devillet, May 21 2019

Number of (2k+1)-ary associative and quasitrivial operations that have two neutral elements 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.

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

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

A307005 Expansion of e.g.f. (2exp(x)-2x-x^2)/(2-2*x-x^2).

Original entry on oeis.org

1, 1, 3, 13, 71, 486, 3982, 38081, 416145, 5116222, 69888746, 1050168417, 17214678241, 305703953660, 5846391071172, 119794781201881, 2618283427770737, 60802908515558346, 1495049717728972990, 38803241993010963977, 1060124286228724147641, 30411290829335509535632

Offset: 0

J. Devillet, Mar 19 2019

Number of totally ordered partitions on an n-element set where each non-minimal class contains at most 2 elements.

Convention a(0) = 1.

Michael De Vlieger, Table of n, a(n) for n = 0..427
Jimmy Devillet, On the single-peakedness property, International summer school "Preferences, decisions and games" (Sorbonne Université, Paris, 2019).
J. Devillet, J.-L. Marichal, and B. Teheux Classifications of quasitrivial semigroups, arXiv:1811.11113 [math.RA], 2018.

Cf. A307006.

Nest[Append[#1, 1 + #2 #1[[-1]] + #2 (#2 - 1) #1[[-2]]/2 ] & @@ {#, Length@ #} &, {1, 1, 3}, 19] (* Michael De Vlieger, Apr 21 2019 *)

my(x='x+O('x^30)); Vec(serlaplace((2*exp(x)-2*x-x^2)/(2-2*x-x^2))) \\ Felix Fröhlich, Mar 19 2019

Recurrence: a(1) = 1, a(2) = 3, a(n+2) = 1 + (n+2)*a(n+1) + (1/2)*(n+2)*(n+1)*a(n).

a(n) = Sum_{i=0..n} (n!/(n + 1 - i)!)*((sqrt(3)/3)*((1 + sqrt(3))/2)^i - (sqrt(3)/3)*((1 - sqrt(3))/2)^i).

More terms from Michel Marcus, Apr 20 2019

A307006 Expansion of e.g.f. (2exp(x)-1-2x-x^2)/(1-x-x^2).

Original entry on oeis.org

1, 1, 4, 20, 130, 1052, 10214, 115684, 1497458, 21806372, 352834942, 6279885284, 121932835754, 2564788969108, 58098821674742, 1410088008633812, 36505125340079074, 1004131069129741124, 29244927598399536878, 899066450011962665092, 29094401487631077315482, 988590340245276942963764

Offset: 0

J. Devillet, Mar 19 2019

Number of associative and quasitrivial binary operations on an n-element set that are order-preserving for some total ordering.

Convention a(0) = 1.

J. Devillet, J.-L. Marichal, and B. Teheux Classifications of quasitrivial semigroups, arXiv:1811.11113 [math.RA], 2018.

Cf. A000045, A307005.

Nest[Append[#1, 2 + #2 #1[[-1]] + #2 (#2 - 1) #1[[-2]] ] & @@ {#, Length@ #} &, {1, 1, 4}, 19] (* Michael De Vlieger, Apr 21 2019 *)
With[{nn=30},CoefficientList[Series[(2Exp[x]-1-2x-x^2)/(1-x-x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 12 2020 *)

my(x='x+O('x^30)); Vec(serlaplace((2*exp(x)-1-2*x-x^2)/(1-x-x^2))) \\ Felix Fröhlich, Mar 19 2019

Recurrence: a(1) = 1, a(2) = 4, a(n+2) = 2 + (n+2)*a(n+1) + (n+2)*(n+1)*a(n).

a(n) = n!*A000045(n) + 2*Sum_{i=0..n} (n!/(n + 1 - i)!)*A000045(i).

More terms from Michel Marcus, Apr 20 2019

A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).

Original entry on oeis.org

0, 1, 2, 6, 14, 30, 62, 126, 254, 510, 1022, 2046, 4094, 8190, 16382, 32766, 65534, 131070, 262142, 524286, 1048574, 2097150, 4194302, 8388606, 16777214, 33554430, 67108862, 134217726, 268435454, 536870910, 1073741822, 2147483646, 4294967294, 8589934590, 17179869182

Offset: 0

J. Devillet, Dec 22 2017

a(n) = A000225(n)-1, a(0)=0, a(1)=1. Number of quasilinear weak orderings R on {1,...,n} that are weakly single-peaked w.r.t. the total ordering 1<...

Essentially the same as A095121 and A000918. - R. J. Mathar, Jan 02 2018

Colin Barker, Table of n, a(n) for n = 0..1000
J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA], 2017-2018.
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A000225, A000918, A095121, A134067.

CoefficientList[Series[x (1 - x + 2 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* or *)
LinearRecurrence[{3, -2}, {0, 1, 2, 6}, 34] (* Michael De Vlieger, Dec 22 2017 *)

concat(0, Vec(x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).
a(n) = 2^n - 2 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3. (End)
a(n) = A134067(n-2) for n >= 3. - Georg Fischer, Oct 30 2018
E.g.f.: 1 + exp(x)*(exp(x) - 2) + x. - Stefano Spezia, May 07 2023

A296964 Expansion of e.g.f. (exp(x)-x)*x/(1-x).

Original entry on oeis.org

0, 1, 2, 9, 40, 205, 1236, 8659, 69280, 623529, 6235300, 68588311, 823059744, 10699776685, 149796873604, 2246953104075, 35951249665216, 611171244308689, 11001082397556420, 209020565553571999, 4180411311071440000, 87788637532500240021, 1931350025715005280484, 44421050591445121451155

Offset: 0

J. Devillet, Dec 22 2017

Number of quasilinear weak orderings R on {1,...,n} and for which {1,...,n} has exactly one maximal element for the quasilinear weak ordering R.

Essentially the same as A038156. - R. J. Mathar, Jan 02 2018

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

Cf. A002627, A038156.

Join[{0,1},Drop[With[{nn=30},CoefficientList[Series[(Exp[x]-x)*x/(1-x),{x,0,nn}],x] Range[0,nn]!],2]] (* Harvey P. Dale, Apr 02 2018 *)

x = QQ[['x']].gen()
f = (exp(x) - x) * x / (1 - x)
f.egf_to_ogf()  # F. Chapoton, Jul 21 2025

a(n) = A002627(n)-1, a(0)=0, a(1)=1.

A296954 Expansion of x(1 - x + 4x^2) / ((1 - x)(1 - 2x)).

Original entry on oeis.org

0, 1, 2, 8, 20, 44, 92, 188, 380, 764, 1532, 3068, 6140, 12284, 24572, 49148, 98300, 196604, 393212, 786428, 1572860, 3145724, 6291452, 12582908, 25165820, 50331644, 100663292, 201326588, 402653180, 805306364, 1610612732, 3221225468, 6442450940, 12884901884

Offset: 0

J. Devillet, Dec 22 2017

Number of bisymmetric, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n} that have annihilator elements.

Apart from the offset the same as A131128. - R. J. Mathar, Jan 02 2018

Colin Barker, Table of n, a(n) for n = 0..1000
J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Cf. A296953.

CoefficientList[Series[x (1 - x + 4 x^2)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* Michael De Vlieger, Dec 23 2017 *)
LinearRecurrence[{3,-2},{0,1,2,8},40] (* Harvey P. Dale, Jun 05 2021 *)

concat(0, Vec(x*(1 - x + 4*x^2) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

a(n) = A296953(n)-2, a(0)=0, a(1)=1.
From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 - x + 4*x^2) / ((1 - x)*(1 - 2*x)).
a(n) = 3*2^(n-1) - 4 for n>1.
a(n) = 3*a(n-1) - 2*a(n-2) for n>3.
(End)

G.f. in the name replaced by a better g.f. by Colin Barker, Dec 23 2017

A296953 Number of bisymmetric, quasitrivial, and order-preserving binary operations on the n-element set {1,...,n}.

Original entry on oeis.org

0, 1, 4, 10, 22, 46, 94, 190, 382, 766, 1534, 3070, 6142, 12286, 24574, 49150, 98302, 196606, 393214, 786430, 1572862, 3145726, 6291454, 12582910, 25165822, 50331646, 100663294, 201326590, 402653182, 805306366, 1610612734, 3221225470, 6442450942, 12884901886

Offset: 0

J. Devillet, Dec 22 2017

Apart from the offset the same as A033484. - R. J. Mathar, Alois P. Heinz, Jan 02 2018

J. Devillet, Bisymmetric and quasitrivial operations: characterizations and enumerations, arXiv:1712.07856 [math.RA] (2017).
Index entries for linear recurrences with constant coefficients, signature (3,-2).

Nest[Append[#, 2 Last@ # + 2] &, {0, 1}, 32] (* or *)
Array[3*2^(# - 1) - 2 + Boole[# == 0]/2 &, 34, 0] (* or *)
CoefficientList[Series[x (1 + x)/((1 - x) (1 - 2 x)), {x, 0, 33}], x] (* Michael De Vlieger, Dec 22 2017 *)

concat(0, Vec(x*(1 + x) / ((1 - x)*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Dec 22 2017

a(0)=0, a(1)=1, a(n+1)-2*a(n) = 2.
From Colin Barker, Dec 22 2017: (Start)
G.f.: x*(1 + x) / ((1 - x)*(1 - 2*x)).
a(n) = 3*2^(n-1) - 2 for n>0.
a(n) = 3*a(n-1) - 2*a(n-2) for n>2.
(End)

G.f. replaced by a better g.f. by Colin Barker, Dec 23 2017

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User: J. Devillet

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

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

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

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A308354 Number of (2k+1)-ary quasitrivial semigroups that have two neutral elements on an n-element set.

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A307005 Expansion of e.g.f. (2*exp(x)-2*x-x^2)/(2-2*x-x^2).

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A307006 Expansion of e.g.f. (2*exp(x)-1-2*x-x^2)/(1-x-x^2).

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A296965 Expansion of x*(1 - x + 2*x^2) / ((1 - x)*(1 - 2*x)).

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A296964 Expansion of e.g.f. (exp(x)-x)*x/(1-x).

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

A307005 Expansion of e.g.f. (2exp(x)-2x-x^2)/(2-2*x-x^2).

A307006 Expansion of e.g.f. (2exp(x)-1-2x-x^2)/(1-x-x^2).

A296965 Expansion of x(1 - x + 2x^2) / ((1 - x)(1 - 2x)).

A296954 Expansion of x(1 - x + 4x^2) / ((1 - x)(1 - 2x)).