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A229936 Sum of all parts of all compositions of n with at least two parts in increasing order.

%I A229936 #19 Nov 01 2013 13:24:20
%S A229936 0,0,0,3,12,45,126,343,848,2034,4700,10648,23652,51935,112798,243120,
%T A229936 520592,1109063,2352366,4971426,10473220,22003464,46115300,96440127,
%U A229936 201288792,419381450,872351896,1811858058,3757992280,7784495839,16105959240,33285784442
%N A229936 Sum of all parts of all compositions of n with at least two parts in increasing order.
%C A229936 Sum of all parts of all compositions of n that are not partitions of n (see example).
%F A229936 a(n) = n*A056823(n) = n*(A011782(n) - A000041(n)).
%F A229936 a(n) = A001787(n) - A066186(n), n >= 1.
%e A229936 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 A229936 ----------------------------------------------------
%e A229936 Compositions       Partitions      Sum of all parts
%e A229936 ----------------------------------------------------
%e A229936 [1, 1, 1, 1]   =   [1, 1, 1, 1]
%e A229936 [2, 1, 1]      =   [2, 1, 1]
%e A229936 [1, 2, 1]                                 4
%e A229936 [3, 1]         =   [3, 1]
%e A229936 [1, 1, 2]                                 4
%e A229936 [2, 2]         =   [2, 2]
%e A229936 [1, 3]                                    4
%e A229936 [4]            =   [4]
%e A229936 ----------------------------------------------------
%e A229936 Total                                    12
%e A229936 .
%e A229936 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 A229936 Cf. A000041, A001787, A011782, A056823, A066186, A229935.
%K A229936 nonn
%O A229936 0,4
%A A229936 _Omar E. Pol_, Oct 14 2013