A007323 -id:A007323 - cp's OEIS frontend

A160254 Expansion of x(2 - 3x + x^2 - 4x^3 + 3x^4 - 2x^5 + x(1 - x - x^3)sqrt((1 + 2x)/(1 - 2x)))/(2(1 - 3x + 3x^2 - 3x^3 + 4x^4 - 3x^5 + 2x^6)).

1, 2, 4, 7, 13, 24, 44, 81, 151, 280, 525, 984, 1859, 3511, 6682, 12709, 24334, 46565, 89626, 172381, 333262, 643733, 1249147, 2421592, 4713715, 9165792, 17888456, 34873456, 68212220, 133269997, 261167821, 511211652, 1003436520, 1967293902

Offset: 1

Jonathan Vos Post, May 06 2009

a(n) is the number of nodes at level n in certain generating tree, denoted C, that embeds the tree of numerical semigroups.

Elizalde (2009) established that the number A007323(n) of numerical semigroups of genus n is bounded in C as follows: A000045(n+2) - 1 <= A007323(n) <= a(n) <= 1 + 3*2^(n - 3).

Matthew House, Table of n, a(n) for n = 1..3328
Sergi Elizalde, Improved bounds on the number of numerical semigroups of a given genus, arXiv:0905.0489 [math.CO], May 4, 2009. See Table 1, p. 8.

Cf. A000045, A007323.

gf : taylor(x*(2 - 3*x + x^2 - 4*x^3 + 3*x^4 - 2*x^5 + x*(1 - x - x^3)*sqrt((1 + 2*x)/(1 - 2*x)))/(2*(1 - 3*x + 3*x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 2*x^6)), x, 0, 100)$
makelist(ratcoef(gf, x, n), n, 1, 100); /* Franck Maminirina Ramaharo, Jan 15 2019 */

Edited, and name replaced by the g.f. by Franck Maminirina Ramaharo, Jan 15 2019

A199711 Triangular array: T(n,k) gives the number of numerical semigroups of genus n and multiplicity k, (n>=1, k>=2).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 3, 6, 7, 5, 1, 1, 3, 7, 10, 11, 6, 1, 1, 3, 9, 13, 17, 16, 7, 1, 1, 4, 11, 16, 27, 28, 22, 8, 1, 1, 4, 13, 22, 37, 44, 44, 29, 9, 1, 1, 4, 15, 24, 49, 64, 72, 66, 37, 10, 1, 1, 5, 18, 32, 66, 85, 116, 116, 95, 46, 11, 1

Offset: 1

Peter Bala, Nov 09 2011

A numerical semigroup is a subset S of N, the nonnegative integers, that is closed under addition, contains the element 0 and such that N-S is finite. The cardinality of N-S is called the genus of S. The least positive integer belonging to S is called the multiplicity of S. The number of numerical semigroups of genus n is A007323(n).

			Triangle begins
.n\k.|..2....3....4....5....6....7....8....9...10
= = = = = = = = = = = = = = = = = = = = = = = = =
..1..|..1
..2..|..1....1
..3..|..1....2....1
..4..|..1....2....3....1
..5..|..1....2....4....4....1
..6..|..1....3....6....7....5....1
..7..|..1....3....7...10...11....6....1
..8..|..1....3....9...13...17...16....7....1
..9..|..1....4...11...16...27...28...22....8....1
...
T(3,3) = 2: The two numerical semigroups of genus 3 and multiplicity 3 are S = N - {1,2,4} and S = N - {1,2,5}.

V. Blanco, P. A. Garcia-Sanchez and J. Puerto, Computing the number of numerical semigroups using generating functions, arXiv:0901.1228v3 [math.CO], 2009.
Nathan Kaplan, Counting Numerical Semigroups, arXiv:1707.02551 [math.CO], 2017.

Cf. A007323 (row sums).

A198896 Number of Cohen-Macaulay modules (see Clark et al. for precise definition).

Original entry on oeis.org

1, 2, 5, 12, 26, 54, 114, 228, 449, 878, 1690

Offset: 0

N. J. A. Sloane, Oct 31 2011

nonn

Sean Clark, Anton Preslicka, Josh Schwartz and Radoslav Zlatev, Some combinatorial conjectures on a family of toric ideals: A report from the MSRI 2011 Commutative Algebra graduate workshop.

a(n) = Sum_{i=0..n} b(i)b(n-i), where b(i) = A007323(i) (with b(0)=1).

A210581 Bras-Amorós number f_n for numerical semigroups of genus n.

Original entry on oeis.org

1, 2, 7, 23, 68, 200, 615, 1764, 5060, 14626, 41785, 117573, 332475, 933891, 2609832, 7278512

Offset: 0

Jonathan Vos Post, Mar 22 2012

M. Bernardini, Fernando Torres, Counting numerical semigroups by genus and even gaps, arXiv:1612.01212 [math.CO], 2016-2017; Discrete Mathematics 340.12 (2017): 2853-2863.
Maria Bras-Amorós, Ordinarization Transform of a Numerical Semigroup and Semigroups with a Large Number of Intervals, arXiv:1203.5000 [math.NT], 2012. See Table on p. 11.

Cf. A007323, A124506.

a(15) from Maria Bras-Amorós, Mar 23 2021

A293176 Irregular triangle read by rows: T(n,k) = number of numerical semigroups of genus n with k even gaps.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 2, 6, 3, 1, 2, 7, 12, 1, 1, 2, 7, 19, 10, 1, 2, 7, 21, 32, 4, 1, 2, 7, 23, 51, 33, 1, 1, 2, 7, 23, 62, 91, 18, 1, 2, 7, 23, 65, 142, 98, 5, 1, 2, 7, 23, 68, 174, 257, 59, 1, 1, 2, 7, 23, 68, 192, 412, 271, 25, 1, 2, 7, 23, 68, 197, 514, 678, 197, 6

Offset: 0

N. J. A. Sloane, Oct 19 2017

			Triangle begins:
1,
1,
1,1,
1,2,1,
1,2,4,
1,2,6,3,
1,2,7,12,1,
1,2,7,19,10,
1,2,7,21,32,4,
1,2,7,23,51,33,1,
1,2,7,23,62,91,18,
1,2,7,23,65,142,98,5,
1,2,7,23,68,174,257,59,1,
1,2,7,23,68,192,412,271,25,
1,2,7,23,68,197,514,678,197,6,
...

M. Bernardini, Fernando Torres. "Counting numerical semigroups by genus and even gaps." Discrete Mathematics 340.12 (2017): 2853-2863. Also arXiv:1612.01212.

Cf. A007323 (row sums).

cp's OEIS Frontend

A160254 Expansion of x(2 - 3x + x^2 - 4x^3 + 3x^4 - 2x^5 + x(1 - x - x^3)sqrt((1 + 2x)/(1 - 2x)))/(2(1 - 3x + 3x^2 - 3x^3 + 4x^4 - 3x^5 + 2x^6)).

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A199711 Triangular array: T(n,k) gives the number of numerical semigroups of genus n and multiplicity k, (n>=1, k>=2).

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A198896 Number of Cohen-Macaulay modules (see Clark et al. for precise definition).

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A210581 Bras-Amorós number f_n for numerical semigroups of genus n.

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A293176 Irregular triangle read by rows: T(n,k) = number of numerical semigroups of genus n with k even gaps.

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cp's OEIS Frontend

A160254 Expansion of x*(2 - 3*x + x^2 - 4*x^3 + 3*x^4 - 2*x^5 + x*(1 - x - x^3)*sqrt((1 + 2*x)/(1 - 2*x)))/(2*(1 - 3*x + 3*x^2 - 3*x^3 + 4*x^4 - 3*x^5 + 2*x^6)).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Maxima

Extensions

A199711 Triangular array: T(n,k) gives the number of numerical semigroups of genus n and multiplicity k, (n>=1, k>=2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A198896 Number of Cohen-Macaulay modules (see Clark et al. for precise definition).

Views

Author

Keywords

References

Formula

A210581 Bras-Amorós number f_n for numerical semigroups of genus n.

Views

Author

Keywords

Links

Crossrefs

Extensions

A293176 Irregular triangle read by rows: T(n,k) = number of numerical semigroups of genus n with k even gaps.

Views

Author

Keywords

Examples

Links

Crossrefs

A160254 Expansion of x(2 - 3x + x^2 - 4x^3 + 3x^4 - 2x^5 + x(1 - x - x^3)sqrt((1 + 2x)/(1 - 2x)))/(2(1 - 3x + 3x^2 - 3x^3 + 4x^4 - 3x^5 + 2x^6)).