OFFSET
0,6
EXAMPLE
The a(4) = 1 through a(9) = 16 partitions:
(211) (311) (411) (322) (422) (522)
(2111) (3111) (511) (611) (711)
(21111) (3211) (4211) (3222)
(4111) (5111) (4221)
(22111) (32111) (4311)
(31111) (41111) (5211)
(211111) (221111) (6111)
(311111) (32211)
(2111111) (33111)
(42111)
(51111)
(321111)
(411111)
(2211111)
(3111111)
(21111111)
For example, the partition y = (4,2,2,1) has mean 9/4 and distinct parts {1,2,4} with mean 7/3, so y is counted under a(9).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Mean[#]<Mean[Union[#]]&]], {n, 0, 30}]
CROSSREFS
These partitions have ranks A360253.
A008284 counts partitions by number of parts.
A116608 counts partitions by number of distinct parts.
A360071 counts partitions by number of parts and number of distinct parts.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 06 2023
STATUS
approved