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A319608 Irregular triangle read by rows: T(n,k) is the number of irreducible numerical semigroups with Frobenius number n and k minimal generators less than n/2. 0
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 5, 2, 1, 4, 1, 1, 4, 2, 1, 4, 2, 1, 7, 6, 1, 1, 4, 2, 1, 8, 9, 2, 1, 5, 4, 1, 1, 7, 8, 2, 1, 8, 9, 2, 1, 10, 17, 7, 1, 1, 5, 6, 2, 1, 10, 19, 12, 2, 1, 10, 16, 7, 1, 1, 10, 21, 11, 2, 1, 9, 16, 9, 2, 1, 13, 34, 26, 8, 1, 1, 8, 15, 10, 2, 1, 14, 41, 37, 14, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,9

COMMENTS

The length of each row is floor((n+1)/2) - floor(n/3).

Summing rows yields A158206.

The expected number of minimal generators of a randomly selected numerical semigroup S(M,p) equals Sum_{n=1..M} ( p * (1 - p)^(floor(n/2)) * Product_{k>=0} T(n,k)*p^k ).

LINKS

Table of n, a(n) for n=1..101.

J. De Loera, C. O'Neill, and D. Wilbourne, Random numerical semigroups and a simplicial complex of irreducible semigroups, arXiv:1710.00979 [math.AC], 2017.

C. Leng and C. O'Neill, A sequence of quasipolynomials arising from random numerical semigroups, arXiv:1809.09915 [math.CO], 2018.

C. O'Neill, The first 90 rows formatted as a triangle

EXAMPLE

T(13,2) = 2, since {5,6,9} and {7,8,9,10,11,12} minimally generate irreducible numerical semigroups with Frobenius number 13.

When written in rows:

  1

  1

  1

  1

  1,  1

  1

  1,  2

  1,  1

  1,  2

  1,  2

  1,  4,  1

  1,  1

  1,  5,  2

  1,  4,  1

  1,  4,  2

  1,  4,  2

  1,  7,  6,  1

  1,  4,  2

  1,  8,  9,  2

  1,  5,  4,  1

  1,  7,  8,  2

  1,  8,  9,  2

  1, 10, 17,  7,  1

  1,  5,  6,  2

  1, 10, 19, 12,  2

  1, 10, 16,  7,  1

  1, 10, 21, 11,  2

  1,  9, 16,  9,  2

  1, 13, 34, 26,  8,  1

  1,  8, 15, 10,  2

CROSSREFS

Cf. A008615, A158206.

Sequence in context: A156775 A064693 A325614 * A230850 A072085 A054868

Adjacent sequences:  A319605 A319606 A319607 * A319609 A319610 A319611

KEYWORD

nonn,tabf

AUTHOR

Christopher O'Neill, Sep 24 2018

STATUS

approved

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Last modified August 2 10:49 EDT 2021. Contains 346422 sequences. (Running on oeis4.)