This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361653 #10 Mar 24 2023 17:01:04
%S A361653 0,0,1,0,3,1,5,3,11,7,17,16,32,31,52,55,90,99,144,167,236,273,371,442,
%T A361653 587,696,901,1078,1379,1651,2074,2489,3102,3707,4571,5467,6692,7982,
%U A361653 9696,11543,13949,16563,19891,23572,28185,33299,39640,46737,55418,65164
%N A361653 Number of even-length integer partitions of n with integer median.
%C A361653 The median of an even-length multiset is the average of the two middle parts.
%C A361653 Because any odd-length partition has integer median, the odd-length version is counted by A027193, strict case A067659.
%e A361653 The a(2) = 1 through a(9) = 7 partitions:
%e A361653 (11) . (22) (2111) (33) (2221) (44) (3222)
%e A361653 (31) (42) (4111) (53) (4221)
%e A361653 (1111) (51) (211111) (62) (4311)
%e A361653 (3111) (71) (6111)
%e A361653 (111111) (2222) (321111)
%e A361653 (3221) (411111)
%e A361653 (3311) (21111111)
%e A361653 (5111)
%e A361653 (221111)
%e A361653 (311111)
%e A361653 (11111111)
%e A361653 For example, the partition (4,3,1,1) has length 4 and median 2, so is counted under a(9).
%t A361653 Table[Length[Select[IntegerPartitions[n], EvenQ[Length[#]]&&IntegerQ[Median[#]]&]],{n,0,30}]
%Y A361653 The odd-length version is counted by A027193, strict A067659.
%Y A361653 Including odd-length partitions gives A307683, complement A325347.
%Y A361653 For mean instead of median we have A361655, any length A067538.
%Y A361653 A000041 counts integer partitions, strict A000009.
%Y A361653 A000975 counts subsets with integer median, mean A051293.
%Y A361653 A359893 and A359901 count partitions by median, odd-length A359902.
%Y A361653 Cf. A008284, A013580, A079309, A240219, A240850, A349156, A359897, A359908, A359912, A360005, A360952.
%K A361653 nonn
%O A361653 0,5
%A A361653 _Gus Wiseman_, Mar 23 2023