Jean-Luc Marichal - cp's OEIS frontend

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

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

A292933 E.g.f.: x/(x+3-2*exp(x)).

Original entry on oeis.org

0, 1, 2, 12, 80, 690, 7092, 85162, 1168400, 18034938, 309307340, 5835250410, 120092842872, 2677545756106, 64289692962068, 1653899162167290, 45384277496827424, 1323216060906107994, 40848835928097158172, 1331096992220322502858

Offset: 0

Jean-Luc Marichal, Sep 27 2017

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

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

Cf. A292932, A292934.

With[{nn=20},CoefficientList[Series[x/(x+3-2Exp[x]),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 01 2019 *)

concat(0, Vec(serlaplace(x/(x+3-2*exp(x))))) \\ Michel Marcus, Sep 27 2017

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

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

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

Original entry on oeis.org

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

A127685 Number of non-isomorphic maximal independent sets of the n-cycle graph.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 5, 5, 7, 6, 10, 9, 14, 14, 20, 20, 30, 31, 44, 48, 67, 74, 104, 117, 161, 188, 254, 302, 407, 489, 654, 801, 1064, 1315, 1742, 2174, 2867, 3613, 4747, 6019, 7900, 10069, 13190, 16895, 22103, 28413, 37150, 47900, 62590, 80912

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

Number of non-isomorphic (i.e. defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph. Also: Number of cyclic compositions of n in which each term is either 2 or 3, where a clockwise writing is not distinguished from its counterclockwise counterpart.

R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, ar2007.

R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:0701647 (2007) and JIS 11 (2008) 08.5.7.

Cf. A127682, A127683, A001608.

a(n) = A127682(n) + Sum(d divides n) A127683(d) = (1/2)*(A127682(n) + (1/n)*(Sum(d divides n) A000010(n/d) A001608(d)))

A127686 Number of non-isomorphic maximal independent sets of the n-cycle graph having no symmetry axis.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 15, 16, 27, 30, 46, 55, 80, 96, 139, 168, 237, 293, 403, 503, 687, 864, 1164, 1477, 1974, 2516, 3348, 4282, 5668, 7284, 9604, 12374, 16279, 21022, 27597, 35718, 46819, 60693, 79480, 103174

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

Number of non-isomorphic (i.e. defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having no symmetry axis. Also: Number of cyclic and non-palindromic compositions of n in which each term is either 2 or 3, where a clockwise writing is not distinguished from its counterclockwise counterpart.

R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:0701647 (2007) and JIS 11 (2008) 08.5.7.

Cf. A127682, A127683, A127685.

a(n) = Sum(d divides n) A127683(d) = A127685(n) - A127682(n)

A127684 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having n isomorphic representatives.

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 3, 3, 2, 5, 5, 6, 7, 9, 7, 13, 12, 16, 15, 24, 19, 32, 28, 44, 37, 58, 46, 81, 63, 104, 86, 145, 110, 189, 151, 257, 200, 339, 260, 460, 351, 599, 464, 813, 610, 1069, 816, 1431, 1078, 1889, 1424, 2530, 1897, 3323, 2513, 4452, 3319

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:0701647 (2007) and JIS 11 (2008) 08.5.7.

Cf. A127682, A127683.

a(n) = (Sum(d divides n) mu(n/d) A127682(d)) + (A127683(n/2)*[n even]).

a(n) = (Sum(d divides n) mu(n/d) A127682(d)) + (A127683(n/2)*[n even])

A127682 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having at least one symmetry axis. Also: Number of cyclic and palindromic compositions of n in which each term is either 2 or 3.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 5, 4, 7, 5, 9, 7, 12, 9, 16, 12, 21, 16, 28, 21, 37, 28, 49, 37, 65, 49, 86, 65, 114, 86, 151, 114, 200, 151, 265, 200, 351, 265, 465, 351, 616, 465, 816, 616, 1081, 816, 1432, 1081, 1897, 1432, 2513, 1897, 3329, 2513, 4410, 3329

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:math/0701647 [math.CO], 2007-2008 and JIS 11 (2008) 08.5.7.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,0,1).

Cf. A000931.

Rest[CoefficientList[Series[-x^2*(x^4+x^3+x^2+x+1)/(x^6+x^4-1),{x,0,63}],x]] (* Vaclav Kotesovec, Mar 29 2014 *)
LinearRecurrence[{0,0,0,1,0,1},{0,1,1,1,1,2},70] (* Harvey P. Dale, Jul 17 2014 *)

concat(0, Vec(-x^2*(x^4+x^3+x^2+x+1)/(x^6+x^4-1) + O(x^100))) \\ Colin Barker, Mar 29 2014

a(n) = A000931(k+3) if n=2k-1 and a(n) = A000931(k+5) if n=2k.
a(n) = a(n-4) + a(n-6).
G.f.: -x^2*(x^4+x^3+x^2+x+1) / (x^6+x^4-1). - Colin Barker, Mar 29 2014

A127687 Number of unlabeled maximal independent sets in the n-cycle graph.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 5, 6, 7, 7, 11, 11, 16, 19, 24, 28, 39, 46, 60, 75, 97, 120, 159, 197, 257, 327, 422, 539, 700, 892, 1157, 1488, 1928, 2479, 3219, 4148, 5383, 6961, 9029, 11687, 15184, 19673, 25564, 33174, 43125, 56010, 72868, 94719, 123283

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

Number of unlabeled (i.e., defined up to a rotation) maximal independent sets in the n-cycle graph. Also: Number of cyclic compositions of n in which each term is either 2 or 3.

(a(n)*n - A001608(n)) mod 2 is a binary sequence of period 14: [0,0,0,0,0,1,0,0,0,1,0,1,0,1]. - Richard Turk, Aug 25 2017

G. C. Greubel, Table of n, a(n) for n = 1..5000
R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:0701647 [math.CO] (2007) and JIS 11 (2008) 08.5.7.
P. Flajolet and M. Soria, The Cycle Construction In SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.

Cf. A001608, A113788, A127682, A127685.

with(numtheory):
perrin:=proc(n) option remember: if n=0 then 3 elif n=1 then 0 elif n=2 then 2 else perrin(n-2)+perrin(n-3) fi end:
a:=proc(n) local d,N:d:=divisors(n);N:=nops(d):
add(phi(n/d[k])*perrin(d[k]),k=1..N)/n end:
seq(a(n),n=1..50);
# Robert FERREOL, Apr 10 2024

(* p = A001608 *) p[n_] := p[n] = p[n - 2] + p[n - 3]; p[0] = 3; p[1] = 0; p[2] = 2; A113788[n_] := (1/n)*Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; a[n_] := Sum[A113788[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jul 16 2012, from formula *)

N = 66;  x = 'x + O('x^N);
f(x) = x^2+x^3;
gf = 'c0 + sum(n=1,N, eulerphi(n)/n*log(1/(1-f(x^n)))  );
v = Vec(gf); v[1]-='c0; v  /* includes a(0)=0 */
/* Joerg Arndt, Jan 21 2013 */

a(n) = Sum_{d|n} A113788(d) = 2 * A127685(n) - A127682(n) = (1/n)*(Sum_{d|n} A000010(n/d) * A001608(d)).

G.f.: Sum_{k>=1} (phi(k)/k) * log( 1/(1-B(x^k)) ) where B(x) = x^2 + x^3. - Joerg Arndt, Jan 21 2013

A127683 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having 2n isomorphic representatives.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 2, 5, 4, 8, 9, 15, 16, 27, 30, 46, 54, 80, 96, 139, 167, 237, 292, 403, 501, 687, 862, 1164, 1472, 1974, 2512, 3347, 4274, 5668, 7275, 9604, 12359, 16278, 21006, 27597, 35690, 46819, 60663, 79478, 103128

Offset: 1

Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007

R. Bisdorff and J.-L. Marichal, Counting non-isomorphic maximal independent sets of the n-cycle graph, arXiv:math/0701647 [math.CO], 2007-2008.

Cf. A113788, A127682.

a(n) = (1/2)*(A113788(n) - Sum_{d|n} mu(n/d)*A127682(d)).

cp's OEIS Frontend

User: Jean-Luc Marichal

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

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A292933 E.g.f.: x/(x+3-2*exp(x)).

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

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A127685 Number of non-isomorphic maximal independent sets of the n-cycle graph.

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A127686 Number of non-isomorphic maximal independent sets of the n-cycle graph having no symmetry axis.

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A127684 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having n isomorphic representatives.

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A127682 Number of non-isomorphic (i.e., defined up to a rotation and a reflection) maximal independent sets of the n-cycle graph having at least one symmetry axis. Also: Number of cyclic and palindromic compositions of n in which each term is either 2 or 3.

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A127687 Number of unlabeled maximal independent sets in the n-cycle graph.

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