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%I #36 Jun 14 2020 12:48:50
%S 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,
%T 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,
%U 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
%N 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.
%C The length of each row is floor((n+1)/2) - floor(n/3).
%C Summing rows yields A158206.
%C 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 ).
%H J. De Loera, C. O'Neill, and D. Wilbourne, <a href="https://arxiv.org/abs/1710.00979">Random numerical semigroups and a simplicial complex of irreducible semigroups</a>, arXiv:1710.00979 [math.AC], 2017.
%H C. Leng and C. O'Neill, <a href="https://arxiv.org/abs/1809.09915">A sequence of quasipolynomials arising from random numerical semigroups</a>, arXiv:1809.09915 [math.CO], 2018.
%H C. O'Neill, <a href="https://gist.github.com/coneill-math/c2f12c94c7ee12ac7652096329417b7d">The first 90 rows formatted as a triangle</a>
%e T(13,2) = 2, since {5,6,9} and {7,8,9,10,11,12} minimally generate irreducible numerical semigroups with Frobenius number 13.
%e When written in rows:
%e 1
%e 1
%e 1
%e 1
%e 1, 1
%e 1
%e 1, 2
%e 1, 1
%e 1, 2
%e 1, 2
%e 1, 4, 1
%e 1, 1
%e 1, 5, 2
%e 1, 4, 1
%e 1, 4, 2
%e 1, 4, 2
%e 1, 7, 6, 1
%e 1, 4, 2
%e 1, 8, 9, 2
%e 1, 5, 4, 1
%e 1, 7, 8, 2
%e 1, 8, 9, 2
%e 1, 10, 17, 7, 1
%e 1, 5, 6, 2
%e 1, 10, 19, 12, 2
%e 1, 10, 16, 7, 1
%e 1, 10, 21, 11, 2
%e 1, 9, 16, 9, 2
%e 1, 13, 34, 26, 8, 1
%e 1, 8, 15, 10, 2
%Y Cf. A008615, A158206.
%K nonn,tabf
%O 1,9
%A _Christopher O'Neill_, Sep 24 2018
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