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 A361906 #8 Apr 01 2023 22:03:36
%S A361906 0,0,0,0,0,2,3,5,9,15,19,36,43,68,96,125,171,232,297,418,529,676,853,
%T A361906 1156,1393,1786,2316,2827,3477,4484,5423,6677,8156,10065,12538,15121,
%U A361906 17978,22091,26666,32363,38176,46640,55137,66895,79589,92621,111485,133485
%N A361906 Number of integer partitions of n such that (length) * (maximum) >= 2*n.
%C A361906 Also partitions such that (maximum) >= 2*(mean).
%C A361906 These are partitions whose complement (see example) has size >= n.
%e A361906 The a(6) = 2 through a(10) = 15 partitions:
%e A361906 (411) (511) (611) (621) (721)
%e A361906 (3111) (4111) (4211) (711) (811)
%e A361906 (31111) (5111) (5211) (5221)
%e A361906 (41111) (6111) (5311)
%e A361906 (311111) (42111) (6211)
%e A361906 (51111) (7111)
%e A361906 (321111) (42211)
%e A361906 (411111) (43111)
%e A361906 (3111111) (52111)
%e A361906 (61111)
%e A361906 (421111)
%e A361906 (511111)
%e A361906 (3211111)
%e A361906 (4111111)
%e A361906 (31111111)
%e A361906 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 A361906 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 A361906 The partition y = (3,2,1,1) has diagram:
%e A361906 o o o
%e A361906 o o .
%e A361906 o . .
%e A361906 o . .
%e A361906 with complement (shown in dots) of size 5, and 5 is not >= 7, so y is not counted under a(7).
%t A361906 Table[Length[Select[IntegerPartitions[n],Length[#]*Max@@#>=2n&]],{n,30}]
%Y A361906 For length instead of mean we have A237752, reverse A237755.
%Y A361906 For minimum instead of mean we have A237821, reverse A237824.
%Y A361906 For median instead of mean we have A361859, reverse A361848.
%Y A361906 The unequal case is A361907.
%Y A361906 The complement is counted by A361852.
%Y A361906 The equal case is A361853, ranks A361855.
%Y A361906 Reversing the inequality gives A361851.
%Y A361906 A000041 counts integer partitions, strict A000009.
%Y A361906 A008284 counts partitions by length, A058398 by mean.
%Y A361906 A051293 counts subsets with integer mean.
%Y A361906 A067538 counts partitions with integer mean, strict A102627, ranks A316413.
%Y A361906 A268192 counts partitions by complement size, ranks A326844.
%Y A361906 Cf. A027193, A111907, A116608, A237984, A324521, A327482, A349156, A360068.
%K A361906 nonn
%O A361906 1,6
%A A361906 _Gus Wiseman_, Mar 29 2023