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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 1 22:36 EDT 2021. Contains 346408 sequences. (Running on oeis4.)