%I #5 Apr 03 2023 09:17:07

%S 0,0,0,1,2,3,7,10,15,23,34,46,67,90,121,164,219,285,375,483,622,799,

%T 1017,1284,1621,2033,2537,3158,3915,4832,5953,7303,8930,10896,13248,

%U 16071,19451,23482,28272,33977,40736,48741,58201,69367,82506,97986,116139

%N Number of integer partitions of n such that the maximum is greater than or equal to twice the median.

%C The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

%e The a(4) = 1 through a(9) = 15 partitions:

%e (211) (311) (411) (421) (422) (522)

%e (2111) (3111) (511) (521) (621)

%e (21111) (3211) (611) (711)

%e (4111) (4211) (4221)

%e (22111) (5111) (4311)

%e (31111) (32111) (5211)

%e (211111) (41111) (6111)

%e (221111) (33111)

%e (311111) (42111)

%e (2111111) (51111)

%e (321111)

%e (411111)

%e (2211111)

%e (3111111)

%e (21111111)

%e The partition y = (5,2,2,1) has maximum 5 and median 2, and 5 >= 2*2, so y is counted under a(10).

%t Table[Length[Select[IntegerPartitions[n],Max@@#>=2*Median[#]&]],{n,30}]

%Y For length instead of median we have A237752.

%Y For minimum instead of median we have A237821.

%Y Reversing the inequality gives A361848.

%Y The equal case is A361849, ranks A361856.

%Y The unequal case is A361857, ranks A361867.

%Y The complement is counted by A361858.

%Y These partitions have ranks A361868.

%Y For mean instead of median we have A361906.

%Y A000041 counts integer partitions, strict A000009.

%Y A000975 counts subsets with integer median.

%Y A325347 counts partitions with integer median, complement A307683.

%Y A359893 and A359901 count partitions by median.

%Y A360005 gives twice median of prime indices, distinct A360457.

%Y Cf. A008284, A027193, A067538, A237755, A237820, A237824, A240219, A359907, A361851, A361860, A361907.

%K nonn

%O 1,5

%A _Gus Wiseman_, Apr 02 2023