A229935 Total number of parts in all compositions of n with at least two parts in increasing order.

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

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(n-1) - 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(4-1) = 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