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A265105
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.
1
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, 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, 1, 2, 0, 4, 0, 2, 2, 1, 0, 2, 0, 2
OFFSET
1,2
COMMENTS
See Dahlberg et al. reference for definition of avoidance and lb.
LINKS
S. Dahlberg, R. Dorward, J. Gerhard, T. Grubb, C. Purcell, L. Reppuhn, B. E. Sagan, Set partition patterns and statistics, arXiv:1502.00056 [math.CO], 2015.
S. Dahlberg, R. Dorward, J. Gerhard, T. Grubb, C. Purcell, L. Reppuhn, B. E. Sagan, Set partition patterns and statistics, Discrete Math., 339 (1): 1-16, 2016.
FORMULA
T(n,k) = #{d>=1: d | k and d+(k/d)+1<=n} + delta_{k,0}, where delta is the Kronecker delta function.
Formula for generating function, fixing n: 1 + sum 1<=m<=n-1, sum 1<=i<=m, q^((n-m)(m-i)).
When k<=n-2, T(n,k) = A000005(k).
EXAMPLE
Triangle begins:
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,
8,1,2,2,3,2,4,0,2,1,2,0,2,
9,1,2,2,3,2,4,2,2,1,2,0,4,0,0,2,1
MATHEMATICA
row[n_] := CoefficientList[1 + Sum[q^((n-m)(m-i)), {m, n-1}, {i, m}], q];
Array[row, 10] // Flatten (* Jean-François Alcover, Sep 26 2018 *)
PROG
(PARI) T(n, k) = if (k==0, n, sumdiv(k, d, (d>=1) && (d+(k/d)+1)<=n));
tabf(nn) = {for (n=1, nn, for (k=0, (n-1)^2\4, print1(T(n, k), ", "); ); print(); ); } \\ Michel Marcus, Apr 07 2016
CROSSREFS
First column is A000027.
Cf. A000005.
Row sum is A000124.
Row length (fixing n, degree of polynomial in k) is A002620.
Sequence in context: A138967 A274913 A330761 * A035612 A199539 A089555
KEYWORD
nonn,tabf
AUTHOR
Robert Dorward, Apr 06 2016
EXTENSIONS
More terms from Jinyuan Wang, Mar 06 2020
STATUS
approved