cp's OEIS Frontend

A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).

%I A237363 #20 Mar 22 2023 12:52:20
%S A237363 1,0,1,1,2,2,6,6,10,13,20,26,39,50,71,87,121,156,208,265,348,440,566,
%T A237363 712,906,1131,1424,1766,2224,2738,3390,4168,5130,6266,7664,9312,11332,
%U A237363 13723,16603,20004,24112,28942,34708,41522,49612,59031,70308,83479,98992
%N A237363 Number of partitions of n for which 2*(number of distinct parts) <= (number of parts).
%C A237363 a(n) + A237365(n) = A000041(n).
%C A237363 Also the number of integer partitions of n whose median difference is 0. For example, the partition (2,2,2,1,1) is counted because its multiset of differences {0,0,0,1} has median 0. - _Gus Wiseman_, Mar 18 2023
%H A237363 Alois P. Heinz, <a href="/A237363/b237363.txt">Table of n, a(n) for n = 0..800</a>
%e A237363 Among the 22 partitions of 8, these qualify:  [5,1,1,1], [4,4], [4,1,1,1,1], [3,3,1,1], [3,1,1,1,1,1], [2,2,2,2], [2,2,2,1,1], [2,2,1,1,1,1], [2,1,1,1,1,1,1], [1,1,1,1,1,1,1,1], and the remaining 12 do not, so that a(8) = 10.
%t A237363 z = 50; t = Map[Length[Select[IntegerPartitions[#], 2*Length[DeleteDuplicates[#]] <= Length[#] &]] &, Range[z]] (*A237363*)
%t A237363 Table[PartitionsP[n] - t[[n]], {n, 1, z}] (*A237365*) (* _Peter J. C. Moses_, Feb 06 2014 *)
%t A237363 Table[Length[Select[IntegerPartitions[n],Median[Differences[#]]==0&]],{n,0,30}] (* _Gus Wiseman_, Mar 18 2023 *)
%Y A237363 These partitions have ranks A361204.
%Y A237363 A000041 counts integer partitions, strict A000009.
%Y A237363 A008284 counts partitions by number of parts, reverse A058398.
%Y A237363 A116608 counts partitions by number of distinct parts.
%Y A237363 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A237363 Comparing twice the number of distinct parts to the number of parts:
%Y A237363               less: A360254, ranks A360558
%Y A237363              equal: A239959, ranks A067801
%Y A237363            greater: A237365, ranks A361393
%Y A237363      less or equal: A237363, ranks A361204
%Y A237363   greater or equal: A361394, ranks A361395
%Y A237363 Cf. A000975, A027193, A114638, A240219, A325347, A360005, A360244.
%K A237363 nonn
%O A237363 0,5
%A A237363 _Clark Kimberling_, Feb 06 2014