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%I #19 Nov 01 2013 13:24:20
%S 0,0,0,3,12,45,126,343,848,2034,4700,10648,23652,51935,112798,243120,
%T 520592,1109063,2352366,4971426,10473220,22003464,46115300,96440127,
%U 201288792,419381450,872351896,1811858058,3757992280,7784495839,16105959240,33285784442
%N Sum of all parts of all compositions of n with at least two parts in increasing order.
%C Sum of all parts of all compositions of n that are not partitions of n (see example).
%F a(n) = n*A056823(n) = n*(A011782(n) - A000041(n)).
%F a(n) = A001787(n) - A066186(n), n >= 1.
%e 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.
%e ----------------------------------------------------
%e Compositions Partitions Sum of all parts
%e ----------------------------------------------------
%e [1, 1, 1, 1] = [1, 1, 1, 1]
%e [2, 1, 1] = [2, 1, 1]
%e [1, 2, 1] 4
%e [3, 1] = [3, 1]
%e [1, 1, 2] 4
%e [2, 2] = [2, 2]
%e [1, 3] 4
%e [4] = [4]
%e ----------------------------------------------------
%e Total 12
%e .
%e 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.
%Y Cf. A000041, A001787, A011782, A056823, A066186, A229935.
%K nonn
%O 0,4
%A _Omar E. Pol_, Oct 14 2013
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