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A361653
Number of even-length integer partitions of n with integer median.
2
0, 0, 1, 0, 3, 1, 5, 3, 11, 7, 17, 16, 32, 31, 52, 55, 90, 99, 144, 167, 236, 273, 371, 442, 587, 696, 901, 1078, 1379, 1651, 2074, 2489, 3102, 3707, 4571, 5467, 6692, 7982, 9696, 11543, 13949, 16563, 19891, 23572, 28185, 33299, 39640, 46737, 55418, 65164
OFFSET
0,5
COMMENTS
The median of an even-length multiset is the average of the two middle parts.
Because any odd-length partition has integer median, the odd-length version is counted by A027193, strict case A067659.
EXAMPLE
The a(2) = 1 through a(9) = 7 partitions:
(11) . (22) (2111) (33) (2221) (44) (3222)
(31) (42) (4111) (53) (4221)
(1111) (51) (211111) (62) (4311)
(3111) (71) (6111)
(111111) (2222) (321111)
(3221) (411111)
(3311) (21111111)
(5111)
(221111)
(311111)
(11111111)
For example, the partition (4,3,1,1) has length 4 and median 2, so is counted under a(9).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&IntegerQ[Median[#]]&]], {n, 0, 30}]
CROSSREFS
The odd-length version is counted by A027193, strict A067659.
Including odd-length partitions gives A307683, complement A325347.
For mean instead of median we have A361655, any length A067538.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median, mean A051293.
A359893 and A359901 count partitions by median, odd-length A359902.
Sequence in context: A112447 A289360 A290212 * A340601 A289891 A289094
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 23 2023
STATUS
approved