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A360682
Number of integer partitions of n of length > 2 whose second differences have median 0.
3
0, 0, 0, 1, 1, 1, 5, 4, 10, 13, 18, 23, 44, 44, 72, 98, 132, 162, 241, 277, 394, 497, 643, 800, 1076, 1287, 1660, 2078, 2604, 3192, 4065, 4892, 6113, 7490, 9166, 11110, 13717, 16429, 20033, 24201, 29143, 34945, 42251, 50219, 60253, 71852, 85503, 101501, 120899
OFFSET
0,7
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(3) = 1 through a(9) = 13 partitions:
(111) (1111) (11111) (222) (22111) (2222) (333)
(321) (31111) (3221) (432)
(2211) (211111) (3311) (531)
(21111) (1111111) (22211) (22221)
(111111) (32111) (33111)
(41111) (51111)
(221111) (222111)
(311111) (321111)
(2111111) (411111)
(11111111) (2211111)
(3111111)
(21111111)
(111111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Median[Differences[#, 2]]==0&]], {n, 0, 30}]
CROSSREFS
For first differences we have A237363.
For sum instead of median we have A360683.
For mean instead of median we have A360683 - A008619.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions with integer median, strict A359907.
A359893 and A359901 count partitions by median, odd-length A359902.
A360005 gives median of prime indices (times two).
Sequence in context: A363323 A309545 A285105 * A089520 A163524 A330614
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 19 2023
STATUS
approved