A229936 Sum of all parts of all compositions of n with at least two parts in increasing order.
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
Offset: 0
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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. ---------------------------------------------------- Compositions Partitions Sum of all parts ---------------------------------------------------- [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] ---------------------------------------------------- Total 12 . 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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