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A234923
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Array w(n,h), in which row n shows the weights, as defined in Comments, of the distinct-parts partitions of n, arranged in Mathematica order.
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1
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1, 2, 3, 5, 4, 7, 5, 9, 8, 6, 11, 10, 14, 7, 13, 12, 11, 17, 8, 15, 14, 13, 20, 19, 9, 17, 16, 15, 23, 14, 22, 20, 10, 19, 18, 17, 26, 16, 25, 24, 23, 30, 11, 21, 20, 19, 29, 18, 28, 17, 27, 26, 25, 34, 12, 23, 22, 21, 32, 20, 31, 19, 30, 29, 29, 28, 38, 26
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OFFSET
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1,2
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COMMENTS
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The weight of a partition P = x (1)+ x(2)+...+x(k) of n is given at A234094 as k*x(1) + (k-1)*x(2) + ... + x(k), which is the number of steps needed to make P from the sum 1+1+...+1 = n by moving dividers into the sum; see the Example section.
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LINKS
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Table of n, a(n) for n=1..68.
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FORMULA
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w(n,h) = dot product of (partition # h of n) and (k, k-1, ..., 1), where k = length of (partition # h of n).
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EXAMPLE
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Represent 1+1+1+1+1 as _1_1_1_1_1_. The partition 3+2+1 matches the placement of dividers d indicated by _1 _1_1d1_1_d_1_d. To place the 1st d takes 3 steps (starting at the 1st '_'); to place the 2nd d takes 3+2 steps (starting at the 1st '_'); to place the 3rd d takes 3+2+1 steps. The total number of steps is 3+5+6 = 14, the 4th number in row 4 because 3+2+1 is the 4th distinct-parts partition of 6 in Mathematica ordering. The first 9 rows:
1
2
3 5
4 7
5 9 8
6 11 10 14
7 13 12 11 17
8 15 14 13 20 19
9 17 16 15 23 14 22 20
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MATHEMATICA
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p[n_] := p[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; q[n_] := q[n] = Length[p[n]]; v[n_] := v[n] = Table[n + 1 - i, {i, 1, n}]; w[n_, h_] := w[n, h] = Dot[p[n][[h]], v[Length[p[n][[h]]]]]; Flatten[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]
TableForm[Table[w[n, h], {n, 1, 9}, {h, 1, q[n]}]]
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CROSSREFS
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Cf. A234094, A234924.
Sequence in context: A085180 A360209 A114750 * A145391 A057756 A340477
Adjacent sequences: A234920 A234921 A234922 * A234924 A234925 A234926
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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Clark Kimberling, Jan 01 2014
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STATUS
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approved
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