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 A362562 #7 Jun 27 2023 19:41:21
%S A362562 0,0,0,0,0,0,0,0,1,0,1,0,4,0,3,3,7,0,12,0,18,12,9,0,52,12,14,33,54,0,
%T A362562 121,0,98,76,31,100,343,0,45,164,493,0,548,0,483,757,88,0,1789,289,
%U A362562 979,645,1290,0,2225,1677,3371,1200,221,0,10649
%N A362562 Number of non-constant integer partitions of n having a unique mode equal to the mean.
%C A362562 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
%e A362562 The a(8) = 1 through a(16) = 7 partitions:
%e A362562 (3221) . (32221) . (4332) . (3222221) (43332) (5443)
%e A362562 (5331) (3322211) (53331) (6442)
%e A362562 (322221) (4222211) (63321) (7441)
%e A362562 (422211) (32222221)
%e A362562 (33222211)
%e A362562 (42222211)
%e A362562 (52222111)
%t A362562 modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
%t A362562 Table[Length[Select[IntegerPartitions[n],!SameQ@@#&&{Mean[#]}==modes[#]&]],{n,0,30}]
%Y A362562 Partitions containing their mean are counted by A237984, ranks A327473.
%Y A362562 Partitions missing their mean are counted by A327472, ranks A327476.
%Y A362562 Allowing constant partitions gives A363723.
%Y A362562 Including median also gives A363728, ranks A363729.
%Y A362562 A000041 counts partitions, strict A000009.
%Y A362562 A008284 counts partitions by length (or decreasing mean), strict A008289.
%Y A362562 A359893 and A359901 count partitions by median.
%Y A362562 A362608 counts partitions with a unique mode.
%Y A362562 Cf. A240219, A325347, A363719, A363720, A363724, A363725, A363731, A363740.
%K A362562 nonn
%O A362562 0,13
%A A362562 _Gus Wiseman_, Jun 27 2023