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A229936
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Sum of all parts of all compositions of n with at least two parts in increasing order.
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1
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0, 0, 0, 3, 12, 45, 126, 343, 848, 2034, 4700, 10648, 23652, 51935, 112798, 243120, 520592, 1109063, 2352366, 4971426, 10473220, 22003464, 46115300, 96440127, 201288792, 419381450, 872351896, 1811858058, 3757992280, 7784495839, 16105959240, 33285784442
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OFFSET
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0,4
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COMMENTS
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Sum of all parts of all compositions of n that are not partitions of n (see example).
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LINKS
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FORMULA
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EXAMPLE
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For n = 4 the table shows both the compositions and the partitions of 4. There are three compositions of 4 that are not partitions of 4.
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Compositions Partitions Sum of all parts
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[1, 1, 1, 1] = [1, 1, 1, 1]
[2, 1, 1] = [2, 1, 1]
[1, 2, 1] 4
[3, 1] = [3, 1]
[1, 1, 2] 4
[2, 2] = [2, 2]
[1, 3] 4
[4] = [4]
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Total 12
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A partition of a positive integer n is any nonincreasing sequence of positive integers which sum to n, ence the compositions of 4 that are not partitions of 4 are [1, 2, 1], [1, 1, 2] and [1, 3]. The sum of all parts of these compositions is 1+3+1+2+1+1+1+2 = 3*4 = 12. On the other hand the sum of all parts in all compositions of 4 is A001787(4) = 32, and the sum of all parts in all partitions of 4 is A066186(4) = 20, so a(4) = 32 - 20 = 12.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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