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Number of compositions of n minus the number of overpartitions of n.
3

%I #13 Feb 17 2014 23:42:42

%S 0,-1,-2,-4,-6,-8,-8,0,28,102,280,680,1544,3368,7152,14912,30706,

%T 62672,127124,256744,516952,1038672,2083864,4176576,8365080,16746150,

%U 33513608,67055456,134148160,268345208,536754288,1073591680,2147291036,4294721040,8589620784

%N Number of compositions of n minus the number of overpartitions of n.

%C Note that a(7) = 0 therefore 7 is the only positive integer whose number of compositions equals the number of overpartitions: A011782(7) = A015128(7) = 64.

%F a(n) = A011782(n) - A015128(n).

%e Illustration of a(4) = -6.

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

%e . Compositions of 4 Overpartitions of 4

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

%e . _ _ _ _ _ _ _ _

%e 1 |_| | | | 1, 1, 1, 1 |.| | | | 1', 1, 1, 1

%e 2 |_ _| | | 2, 1, 1 |_| | | | 1, 1, 1, 1

%e 3 |_| | | 1, 2, 1 | .|.| | 2', 1', 1

%e 4 |_ _ _| | 3, 1 | |.| | 2, 1', 1

%e 5 |_| | | 1, 1, 2 | .| | | 2', 1, 1

%e 6 |_ _| | 2, 2 |_ _| | | 2, 1, 1

%e 7 |_| | 1, 3 | .|.| 3', 1

%e 8 |_ _ _ _| 4 | |.| 3, 1

%e 9 | .| | 3', 1

%e 10 |_ _ _| | 3, 1

%e 11 | .| | 2', 2

%e 12 |_ _| | 2, 2

%e 13 | .| 4'

%e 14 |_ _ _ _| 4

%e .

%e There are 8 compositions of 4 and there are 14 overpartitions of 4, so a(4) = 8 - 14 = -6.

%Y Cf. A011782, A015128, A056823, A230441, A236633, A237044, A237045.

%K sign

%O 0,3

%A _Omar E. Pol_, Feb 02 2014