%I #5 Feb 06 2023 10:06:16
%S 1,1,2,3,4,4,8,6,11,11,17,13,28,19,32,40,48,39,71,55,103,105,110,105,
%T 197,170,195,237,319,257,462,341,515,543,584,784,1028,761,973,1153,
%U 1606,1261,2137,1611,2368,2815,2575,2591,4393,3798,4602,4663,5777,5121
%N Number of integer partitions of n where the parts have the same mean as the distinct parts.
%e The a(1) = 1 through a(8) = 11 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (11111) (51) (61) (62)
%e (222) (421) (71)
%e (321) (1111111) (431)
%e (2211) (521)
%e (111111) (2222)
%e (3221)
%e (3311)
%e (11111111)
%t Table[Length[Select[IntegerPartitions[n],Mean[#]==Mean[Union[#]]&]],{n,0,30}]
%Y For multiplicities instead of distinct parts we have A360068.
%Y The complement is counted by A360242, ranks A360246.
%Y For median instead of mean we have A360245, complement A360244.
%Y These partitions have ranks A360247.
%Y Cf. A360250 and A360251, ranks A360252 and A360253.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by number of parts.
%Y A058398 counts partitions by mean, also A327482.
%Y A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A116608 counts partitions by number of distinct parts.
%Y A360071 counts partitions by number of parts and number of distinct parts.
%Y A360241 counts partitions whose distinct parts have integer mean.
%Y Cf. A051293, A067340, A240219, A316313, A326567/A326568, A326619/A326620, A326621, A349156, A360069.
%K nonn
%O 0,3
%A _Gus Wiseman_, Feb 04 2023