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%I #8 Apr 01 2023 22:03:36
%S 0,0,0,0,0,2,3,5,9,15,19,36,43,68,96,125,171,232,297,418,529,676,853,
%T 1156,1393,1786,2316,2827,3477,4484,5423,6677,8156,10065,12538,15121,
%U 17978,22091,26666,32363,38176,46640,55137,66895,79589,92621,111485,133485
%N Number of integer partitions of n such that (length) * (maximum) >= 2*n.
%C Also partitions such that (maximum) >= 2*(mean).
%C These are partitions whose complement (see example) has size >= n.
%e The a(6) = 2 through a(10) = 15 partitions:
%e (411) (511) (611) (621) (721)
%e (3111) (4111) (4211) (711) (811)
%e (31111) (5111) (5211) (5221)
%e (41111) (6111) (5311)
%e (311111) (42111) (6211)
%e (51111) (7111)
%e (321111) (42211)
%e (411111) (43111)
%e (3111111) (52111)
%e (61111)
%e (421111)
%e (511111)
%e (3211111)
%e (4111111)
%e (31111111)
%e The partition y = (4,2,1,1) has length 4 and maximum 4, and 4*4 >= 2*8, so y is counted under a(8).
%e The partition y = (3,2,1,1) has length 4 and maximum 3, and 4*3 is not >= 2*7, so y is not counted under a(7).
%e The partition y = (3,2,1,1) has diagram:
%e o o o
%e o o .
%e o . .
%e o . .
%e with complement (shown in dots) of size 5, and 5 is not >= 7, so y is not counted under a(7).
%t Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#>=2n&]],{n,30}]
%Y For length instead of mean we have A237752, reverse A237755.
%Y For minimum instead of mean we have A237821, reverse A237824.
%Y For median instead of mean we have A361859, reverse A361848.
%Y The unequal case is A361907.
%Y The complement is counted by A361852.
%Y The equal case is A361853, ranks A361855.
%Y Reversing the inequality gives A361851.
%Y A000041 counts integer partitions, strict A000009.
%Y A008284 counts partitions by length, A058398 by mean.
%Y A051293 counts subsets with integer mean.
%Y A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A268192 counts partitions by complement size, ranks A326844.
%Y Cf. A027193, A111907, A116608, A237984, A324521, A327482, A349156, A360068.
%K nonn
%O 1,6
%A _Gus Wiseman_, Mar 29 2023