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

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Last modified June 19 03:31 EDT 2021. Contains 345125 sequences. (Running on oeis4.)