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A360688
Number of integer partitions of n with integer median of 0-appended first differences.
11
1, 1, 3, 4, 5, 7, 12, 18, 25, 32, 46, 62, 79, 109, 142, 189, 240, 322, 405, 522, 671, 853, 1053, 1345, 1653, 2081, 2551, 3174, 3878, 4826, 5851, 7219, 8747, 10712, 12936, 15719, 18876, 22872, 27365, 32926, 39253, 47070, 55857, 66676, 79029, 93864, 110832
OFFSET
1,3
COMMENTS
Includes all partitions of odd length (A027193).
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(1) = 1 through a(8) = 18 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (22) (41) (42) (43) (44)
(111) (211) (221) (222) (61) (62)
(1111) (311) (321) (322) (332)
(11111) (411) (331) (422)
(21111) (421) (431)
(111111) (511) (521)
(3211) (611)
(22111) (2222)
(31111) (3221)
(211111) (4211)
(1111111) (22211)
(32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
For example, the partition y = (3,2,2,1) has 0-appended parts (3,2,2,1,0), with differences (1,0,1,1), and the multiset {0,1,1,1} has median 1, so y is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], IntegerQ[Median[Differences[Prepend[Reverse[#], 0]]]]&]], {n, 30}]
CROSSREFS
The case of median 0 is A360254, ranks A360558.
These partitions have ranks A360556, complement A360557.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by number of parts.
A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Sequence in context: A120424 A139440 A102607 * A079463 A325422 A190213
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 20 2023
STATUS
approved