

A229936


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


1



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
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OFFSET

0,4


COMMENTS

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


LINKS

Table of n, a(n) for n=0..31.


FORMULA

a(n) = n*A056823(n) = n*(A011782(n)  A000041(n)).
a(n) = A001787(n)  A066186(n), n >= 1.


EXAMPLE

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.
Sequence in context: A005656 A339066 A260146 * A258626 A064017 A005320
Adjacent sequences: A229933 A229934 A229935 * A229937 A229938 A229939


KEYWORD

nonn


AUTHOR

Omar E. Pol, Oct 14 2013


STATUS

approved



