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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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See Dahlberg et al. reference for definition of avoidance and lb.

LINKS

Table of n, a(n) for n=1..59.

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

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: A135560 A138967 A274913 * A035612 A199539 A089555

Adjacent sequences:  A265102 A265103 A265104 * A265106 A265107 A265108

KEYWORD

nonn,tabf

AUTHOR

Robert Dorward, Apr 06 2016

STATUS

approved

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Last modified February 24 03:49 EST 2018. Contains 299595 sequences. (Running on oeis4.)