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A229936 Sum of all parts of all compositions of n with at least two parts in increasing order. 1

%I

%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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Last modified July 28 23:26 EDT 2021. Contains 346340 sequences. (Running on oeis4.)