|
|
A238478
|
|
Number of partitions of n whose median is a part.
|
|
18
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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 not is 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
|
|
|
FORMULA
|
For all n, a(n) >= A027193(n) [Because when a partition of n has an odd number of parts, then its median is simply the part at the middle] - Antti Karttunen, Feb 27 2014
|
|
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
|
These partitions have ranks A362618.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|