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%I #33 Mar 06 2020 08:10:20
%S 1,2,3,1,4,1,2,5,1,2,2,1,6,1,2,2,3,0,2,7,1,2,2,3,2,2,0,2,1,8,1,2,2,3,
%T 2,4,0,2,1,2,0,2,9,1,2,2,3,2,4,2,2,1,2,0,4,0,0,2,1,10,1,2,2,3,2,4,2,4,
%U 1,2,0,4,0,2,2,1,0,2,0,2
%N Triangle T(n,k) of coefficients of q^k in LB_n(12/3), set partitions that avoid 12/3 with lb=k. Related to a restricted divisor function.
%C See Dahlberg et al. reference for definition of avoidance and lb.
%H Jinyuan Wang, <a href="/A265105/b265105.txt">Rows n = 1..50 of triangle, flattened</a>
%H S. Dahlberg, R. Dorward, J. Gerhard, T. Grubb, C. Purcell, L. Reppuhn, B. E. Sagan, <a href="https://arxiv.org/abs/1502.00056">Set partition patterns and statistics</a>, arXiv:1502.00056 [math.CO], 2015.
%H S. Dahlberg, R. Dorward, J. Gerhard, T. Grubb, C. Purcell, L. Reppuhn, B. E. Sagan, <a href="http://dx.doi.org/10.1016/j.disc.2015.07.001">Set partition patterns and statistics</a>, Discrete Math., 339 (1): 1-16, 2016.
%F T(n,k) = #{d>=1: d | k and d+(k/d)+1<=n} + delta_{k,0}, where delta is the Kronecker delta function.
%F Formula for generating function, fixing n: 1 + sum 1<=m<=n-1, sum 1<=i<=m, q^((n-m)(m-i)).
%F When k<=n-2, T(n,k) = A000005(k).
%e Triangle begins:
%e 1,
%e 2,
%e 3,1,
%e 4,1,2,
%e 5,1,2,2,1,
%e 6,1,2,2,3,0,2,
%e 7,1,2,2,3,2,2,0,2,1,
%e 8,1,2,2,3,2,4,0,2,1,2,0,2,
%e 9,1,2,2,3,2,4,2,2,1,2,0,4,0,0,2,1
%t row[n_] := CoefficientList[1 + Sum[q^((n-m)(m-i)), {m, n-1}, {i, m}], q];
%t Array[row, 10] // Flatten (* _Jean-François Alcover_, Sep 26 2018 *)
%o (PARI) T(n, k) = if (k==0, n, sumdiv(k, d, (d>=1) && (d+(k/d)+1)<=n));
%o tabf(nn) = {for (n=1, nn, for (k=0, (n-1)^2\4, print1(T(n, k), ", ");); print(););} \\ _Michel Marcus_, Apr 07 2016
%Y First column is A000027.
%Y Cf. A000005.
%Y Row sum is A000124.
%Y Row length (fixing n, degree of polynomial in k) is A002620.
%K nonn,tabf
%O 1,2
%A _Robert Dorward_, Apr 06 2016
%E More terms from _Jinyuan Wang_, Mar 06 2020
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