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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: A325614 A361746 A369917 * 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 September 17 04:04 EDT 2024. Contains 375984 sequences. (Running on oeis4.)