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

Author

Omar E. Pol, Oct 14 2013

Keywords

• nonn

Comments

Sum of all parts of all compositions of n that are not partitions of n (see example).

Examples

			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.
		

Crossrefs

Cf. A000041, A001787, A011782, A056823, A066186, A229935.

Formula

a(n) = n*A056823(n) = n*(A011782(n) - A000041(n)).

a(n) = A001787(n) - A066186(n), n >= 1.