cp's OEIS Frontend

A238478 Number of partitions of n whose median is a part.

%I A238478 #31 Apr 25 2024 05:55:36
%S A238478 1,2,2,4,5,8,11,17,22,32,43,59,78,105,136,181,233,302,386,496,626,796,
%T A238478 999,1255,1564,1951,2412,2988,3674,4516,5524,6753,8211,9984,12086,
%U A238478 14617,17617,21211,25450,30514,36475,43550,51869,61707,73230,86821,102706
%N A238478 Number of partitions of n whose median is a part.
%C A238478 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
%F A238478 a(n) + A238479(n) = A000041(n).
%F A238478 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
%F A238478 a(n) = A078408(n-1) - A282893(n). - _Mathew Englander_, May 24 2023
%e A238478 a(6) counts these partitions:  6, 411, 33, 321, 3111, 222, 21111, 111111.
%t A238478 Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Median[p]]], {n, 40}]
%Y A238478 For mean instead of median we have A237984, ranks A327473.
%Y A238478 The complement is counted by A238479, ranks A362617.
%Y A238478 These partitions have ranks A362618.
%Y A238478 A000041 counts integer partitions.
%Y A238478 A325347 counts partitions with integer median, complement A307683.
%Y A238478 A359893/A359901/A359902 count partitions by median.
%Y A238478 A359908 ranks partitions with integer median, complement A359912.
%Y A238478 Cf. A002865, A008284, A053263, A238480, A240219, A327472, A359907, A360686, A360687, A362608.
%K A238478 nonn,easy
%O A238478 1,2
%A A238478 _Clark Kimberling_, Feb 27 2014