

A229935


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


1



0, 0, 0, 2, 8, 28, 77, 202, 490, 1152, 2624, 5869, 12913, 28116, 60660, 130004, 277065, 587859, 1242540, 2617942, 5500394, 11528284, 24109349, 50321442, 104844426, 218086957, 452963310, 939496802, 1946122511, 4026488387, 8321444573, 17179801049, 35433395265
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OFFSET

0,4


COMMENTS

Total number of parts in all compositions of n that are not partitions of n (see example).


LINKS

Table of n, a(n) for n=0..32.


FORMULA

a(n) = A001792(n1)  A006128(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 Number of parts

[1, 1, 1, 1] = [1, 1, 1, 1]
[2, 1, 1] = [2, 1, 1]
[1, 2, 1] 3
[3, 1] = [3, 1]
[1, 1, 2] 3
[2, 2] = [2, 2]
[1, 3] 2
[4] = [4]

Total 8
.
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 total number of parts of these compositions is 3 + 3 + 2 = 8. On the other hand the total number of parts in all compositions of 4 is A001792(41) = 20, and the total number of parts in all partitions of 4 is A006128(4) = 12, so a(4) = 20  12 = 8.


CROSSREFS

Cf. A000041, A001792, A006128, A001782, A056823, A229936.
Sequence in context: A197932 A170871 A245996 * A082107 A135263 A048497
Adjacent sequences: A229932 A229933 A229934 * A229936 A229937 A229938


KEYWORD

nonn


AUTHOR

Omar E. Pol, Oct 14 2013


STATUS

approved



