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Total number of parts in all compositions of n with at least two parts in increasing order.
1

%I #23 Nov 01 2013 13:24:48

%S 0,0,0,2,8,28,77,202,490,1152,2624,5869,12913,28116,60660,130004,

%T 277065,587859,1242540,2617942,5500394,11528284,24109349,50321442,

%U 104844426,218086957,452963310,939496802,1946122511,4026488387,8321444573,17179801049,35433395265

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

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

%F a(n) = A001792(n-1) - A006128(n), n >= 1.

%e 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.

%e ----------------------------------------------------

%e Compositions Partitions Number of parts

%e ----------------------------------------------------

%e [1, 1, 1, 1] = [1, 1, 1, 1]

%e [2, 1, 1] = [2, 1, 1]

%e [1, 2, 1] 3

%e [3, 1] = [3, 1]

%e [1, 1, 2] 3

%e [2, 2] = [2, 2]

%e [1, 3] 2

%e [4] = [4]

%e ----------------------------------------------------

%e Total 8

%e .

%e 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.

%Y Cf. A000041, A001792, A006128, A001782, A056823, A229936.

%K nonn

%O 0,4

%A _Omar E. Pol_, Oct 14 2013