A360682 Number of integer partitions of n of length > 2 whose second differences have median 0.

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).

Links

Table of n, a(n) for n=0..48.

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).

Cf. A000975, A027193, A067538, A114638, A307683, A359908, A360245, A360254, A360687, A360688.

Sequence in context: A363323 A309545 A285105 * A089520 A163524 A330614

Adjacent sequences: A360679 A360680 A360681 * A360683 A360684 A360685

Keyword

nonn

Author

Gus Wiseman, Feb 19 2023

Status

approved