login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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

%I #22 Feb 17 2014 23:42:28

%S 0,0,0,4,12,36,104,260,628,1448,3344,7464,16564,36180,78480,169232,

%T 362732,774172,1645508,3485788,7360208,15503432,32571360,68289536,

%U 142880552,298417848,622194236,1295266596,2692514348,5589496748,11588789220,23998902548

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

%C Number of overcompositions of n that contain at least two parts in increasing order.

%F a(n) = A236002(n) - A015128(n).

%e Illustration of a(4) = -6 with both overcompositions and overpartitions in colexicographic order.

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

%e . Overcompositions of 4 Overpartitions of 4

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

%e . _ _ _ _ _ _ _ _

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

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

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

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

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

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

%e 7 *|.| .| | 1', 2', 1 | .|.| 3', 1

%e 8 *| | .| | 1, 2', 1 | |.| 3, 1

%e 9 *|.| | | 1', 2, 1 | .| | 3', 1

%e 10 *|_| | | 1, 2, 1 |_ _ _| | 3, 1

%e 11 | .|.| 3', 1' | .| | 2', 2

%e 12 | |.| 3, 1' |_ _| | 2, 2

%e 13 | .| | 3', 1 | .| 4'

%e 14 |_ _ _| | 3, 1 |_ _ _ _| 4

%e 15 *|.| | .| 1', 1, 2'

%e 16 *| | | .| 1, 1, 2'

%e 17 *|.| | | 1', 1, 2

%e 18 *|_| | | 1, 1, 2

%e 19 | .| | 2', 2

%e 20 |_ _| | 2, 2

%e 21 *|.| .| 1', 3'

%e 22 *| | .| 1, 3'

%e 23 *|.| | 1', 3

%e 24 *|_| | 1, 3

%e 25 | .| 4'

%e 26 |_ _ _ _| 4

%e .

%e There are 26 overcompositions of 4 and there are 14 overpartitions of 4, so the difference is a(4) = 26 - 14 = 12.

%e On the other hand there are 12 overcompositions of 4 that contain at least two parts in increasing order, so a(4) = 12.

%Y Cf. A000041, A011782, A015128, A056823, A230441, A236002, A236633, A237044, A237047, A237272.

%K nonn

%O 0,4

%A _Omar E. Pol_, Feb 02 2014