%I #8 Jun 25 2023 18:20:17

%S 0,0,0,0,0,0,1,0,0,2,1,0,5,0,1,8,5,0,12,0,19,14,2,0,52,21,3,23,59,0,

%T 122,0,97,46,6,167,303,0,8,82,559,0,543,0,355,745,15,0,1685,510,1083,

%U 251,840,0,2325,1832,3692,426,34,0,9599

%N Number of integer partitions of n whose mean is a mode but not the only mode.

%C 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 The a(n) partitions for n = 6, 9, 12, 15, 18:

%e (3,2,1) (4,3,2) (5,4,3) (6,5,4) (7,6,5)

%e (5,3,1) (6,4,2) (7,5,3) (8,6,4)

%e (7,4,1) (8,5,2) (9,6,3)

%e (6,3,2,1) (9,5,1) (10,6,2)

%e (3,3,2,2,1,1) (4,4,3,3,1) (11,6,1)

%e (5,3,3,2,2) (4,4,3,3,2,2)

%e (5,4,3,2,1) (5,5,3,3,1,1)

%e (7,3,3,1,1) (6,4,3,3,1,1)

%e (7,3,3,2,2,1)

%e (8,3,3,2,1,1)

%e (3,3,3,2,2,2,1,1,1)

%e (6,2,2,2,2,1,1,1,1)

%t modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];

%t Table[Length[Select[IntegerPartitions[n],MemberQ[modes[#],Mean[#]]&&!{Mean[#]}==modes[#]&]],{n,30}]

%Y For a unique mode we have A363723, non-constant A362562.

%Y For any number of modes we have A363724.

%Y A000041 counts partitions, strict A000009.

%Y A008284 counts partitions by length (or decreasing mean), strict A008289.

%Y A237984 counts partitions containing their mean, ranks A327473.

%Y A327472 counts partitions not containing their mean, ranks A327476.

%Y A362608 counts partitions with a unique mode, ranks A356862.

%Y A363719 counts partitions with all three averages equal, ranks A363727.

%Y Cf. A240219, A326567/A326568, A359893, A363720, A363725, A363740.

%K nonn

%O 0,10

%A _Gus Wiseman_, Jun 24 2023