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A265105
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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.
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1
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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
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OFFSET
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1,2
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COMMENTS
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See Dahlberg et al. reference for definition of avoidance and lb.
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LINKS
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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.
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FORMULA
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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)).
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EXAMPLE
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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
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MATHEMATICA
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row[n_] := CoefficientList[1 + Sum[q^((n-m)(m-i)), {m, n-1}, {i, m}], q];
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PROG
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(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
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CROSSREFS
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Row length (fixing n, degree of polynomial in k) is A002620.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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