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%I A362048 #5 Apr 11 2023 08:39:46
%S A362048 1,2,2,3,4,6,8,12,15,20,25,33,41,53,66,85,105,134,164,205,250,308,373,
%T A362048 456,549,666,799,963,1152,1382,1645,1965,2330,2767,3269,3865,4546,
%U A362048 5353,6274,7357,8596,10046,11700,13632,15834,18394,21312,24690,28534,32974
%N A362048 Number of integer partitions of n such that (length) <= 2*(median).
%C A362048 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 A362048 The a(1) = 1 through a(9) = 15 partitions:
%e A362048   (1)  (2)   (3)   (4)   (5)    (6)    (7)     (8)     (9)
%e A362048        (11)  (21)  (22)  (32)   (33)   (43)    (44)    (54)
%e A362048                    (31)  (41)   (42)   (52)    (53)    (63)
%e A362048                          (221)  (51)   (61)    (62)    (72)
%e A362048                                 (222)  (322)   (71)    (81)
%e A362048                                 (321)  (331)   (332)   (333)
%e A362048                                        (421)   (422)   (432)
%e A362048                                        (2221)  (431)   (441)
%e A362048                                                (521)   (522)
%e A362048                                                (2222)  (531)
%e A362048                                                (3221)  (621)
%e A362048                                                (3311)  (3222)
%e A362048                                                        (3321)
%e A362048                                                        (4221)
%e A362048                                                        (4311)
%t A362048 Table[Length[Select[IntegerPartitions[n],Length[#]<=2*Median[#]&]],{n,30}]
%Y A362048 For maximum instead of median we have A237755.
%Y A362048 For minimum instead of median we have A237800.
%Y A362048 For maximum instead of length we have A361848.
%Y A362048 The equal case is A362049.
%Y A362048 A000041 counts integer partitions, strict A000009.
%Y A362048 A000975 counts subsets with integer median.
%Y A362048 A325347 counts partitions with integer median, complement A307683.
%Y A362048 A359893 and A359901 count partitions by median.
%Y A362048 A360005 gives twice median of prime indices, distinct A360457.
%Y A362048 Cf. A008284, A013580, A027193, A237824, A240219, A361394, A361851, A361856-A361860, A361867, A361868.
%K A362048 nonn
%O A362048 1,2
%A A362048 _Gus Wiseman_, Apr 10 2023

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