A238478 - OEIS

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A238478

Number of partitions of n whose median is a part.

1, 2, 2, 4, 5, 8, 11, 17, 22, 32, 43, 59, 78, 105, 136, 181, 233, 302, 386, 496, 626, 796, 999, 1255, 1564, 1951, 2412, 2988, 3674, 4516, 5524, 6753, 8211, 9984, 12086, 14617, 17617, 21211, 25450, 30514, 36475, 43550, 51869, 61707, 73230, 86821, 102706

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OFFSET

1,2

COMMENTS

Also the number of integer partitions of n with a unique middle part. This means that either the length is odd or the two middle parts are equal. For example, the partition (4,3,2,1) has middle parts {2,3} so is not counted under a(10), but (3,2,2,1) has middle parts {2,2} so is counted under a(8). - Gus Wiseman, May 13 2023

LINKS

Table of n, a(n) for n=1..47.

FORMULA

a(n) + A238479(n) = A000041(n).

For all n, a(n) >= A027193(n) (because when a partition of n has an odd number of parts, its median is simply the part at the middle). - Antti Karttunen, Feb 27 2014

a(n) = A078408(n-1) - A282893(n). - Mathew Englander, May 24 2023

EXAMPLE

a(6) counts these partitions: 6, 411, 33, 321, 3111, 222, 21111, 111111.

MATHEMATICA

Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Median[p]]], {n, 40}]

CROSSREFS

For mean instead of median we have A237984, ranks A327473.

The complement is counted by A238479, ranks A362617.

These partitions have ranks A362618.

A000041 counts integer partitions.

A325347 counts partitions with integer median, complement A307683.

A359893/A359901/A359902 count partitions by median.