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%I #8 Feb 11 2023 08:12:56
%S 1,1,0,0,2,1,1,0,2,2,5,8,10,14,20,19,26,31,35,41,55,65,85,102,118,151,
%T 181,201,236,281,313,365,424,495,593,688,825,978,1181,1374,1650,1948,
%U 2323,2682,3175,3680,4314,4930,5718,6546,7532,8557,9777,11067,12622
%N Number of integer partitions of n for which the distinct parts have the same median as the multiplicities.
%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(1) = 1 through a(11) = 8 partitions:
%e 1 . . 22 221 3111 . 3311 333 3331 32222
%e 211 41111 32211 33211 33221
%e 42211 44111
%e 322111 52211
%e 511111 322211
%e 332111
%e 422111
%e 3221111
%t Table[Length[Select[IntegerPartitions[n], Median[Length/@Split[#]]==Median[Union[#]]&]],{n,0,30}]
%Y For mean instead of median: A114638, ranks A324570.
%Y For parts instead of multiplicities: A360245, ranks A360249.
%Y These partitions have ranks A360453.
%Y For parts instead of distinct parts: A360456, ranks A360454.
%Y A000041 counts integer partitions, strict A000009.
%Y A116608 counts partitions by number of distinct parts.
%Y A325347 counts partitions w/ integer median, strict A359907, ranks A359908.
%Y A359893 and A359901 count partitions by median, odd-length A359902.
%Y Cf. A000975, A027193, A240219, A326619/A326620, A359894, A359895, A360068, A360243, A360244, A360248.
%K nonn
%O 0,5
%A _Gus Wiseman_, Feb 10 2023
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