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A231147
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Array of coefficients of numerator polynomials of the rational function p(n, x + 1/x), where p(n,x) = (x^n - 1)/(x - 1).
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14
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1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 4, 1, 1, 1, 1, 5, 4, 9, 4, 5, 1, 1, 1, 1, 6, 5, 14, 9, 14, 5, 6, 1, 1, 1, 1, 7, 6, 20, 14, 29, 14, 20, 6, 7, 1, 1, 1, 1, 8, 7, 27, 20, 49, 29, 49, 20, 27, 7, 8, 1, 1, 1, 1, 9, 8, 35, 27, 76, 49, 99, 49, 76, 27, 35, 8, 9
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OFFSET
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1,7
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COMMENTS
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Also appears to be the number of nonempty subsets of {1,...,n} with median k, where k ranges from 1 to n in steps of 1/2, and the median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length). For example, row n = 5 counts the following subsets:
{1} {1,2} {2} {1,4} {3} {2,5} {4} {4,5} {5}
{1,3} {2,3} {1,5} {3,4} {3,5}
{1,2,3} {1,2,3,4} {2,4} {1,3,4,5} {1,4,5}
{1,2,4} {1,2,3,5} {1,3,4} {2,3,4,5} {2,4,5}
{1,2,5} {1,3,5} {3,4,5}
{2,3,4}
{2,3,5}
{1,2,4,5}
{1,2,3,4,5}
Central diagonals T(n,(n+1)/2) appear to be A100066 (bisection A006134).
For mean instead of median we have A327481.
For partitions instead of subsets we have A359893, full steps A359901.
(End)
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LINKS
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EXAMPLE
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Triangle begins:
1
1 1 1
1 1 3 1 1
1 1 4 3 4 1 1
1 1 5 4 9 4 5 1 1
1 1 6 5 14 9 14 5 6 1 1
1 1 7 6 20 14 29 14 20 6 7 1 1
1 1 8 7 27 20 49 29 49 20 27 7 8 1 1
1 1 9 8 35 27 76 49 99 49 76 27 35 8 9 1 1
First 3 polynomials: 1, 1 + x + x^2, 1 + x + 3*x^2 + x^3 + x^4
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MATHEMATICA
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z = 60; p[n_, x_] := p[x] = (x^n - 1)/(x - 1); Table[p[n, x], {n, 1, z/4}]; f1[n_, x_] := f1[n, x] = Numerator[Factor[p[n, x] /. x -> x + 1/x]]; Table[Expand[f1[n, x]], {n, 0, z/4}]
Flatten[Table[CoefficientList[f1[n, x], x], {n, 1, z/4}]]
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CROSSREFS
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Row lengths are 2n-1 = A005408(n-1).
Removing every other column appears to give A013580.
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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STATUS
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approved
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