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, 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

Jinyuan Wang, Rows n = 1..50 of triangle, flattened

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

Adjacent sequences: A265102 A265103 A265104 * A265106 A265107 A265108

Keyword

nonn,tabf

Author

Robert Dorward, Apr 06 2016

Extensions

More terms from Jinyuan Wang, Mar 06 2020

Status

approved