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
KEYWORD
nonn,tabf
AUTHOR
Robert Dorward, Apr 06 2016
EXTENSIONS
More terms from Jinyuan Wang, Mar 06 2020
STATUS
approved