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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 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: 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 October 22 20:57 EDT 2018. Contains 316502 sequences. (Running on oeis4.)