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A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.

%I A364059 #12 Jul 08 2023 23:06:19
%S A364059 0,0,1,2,3,5,9,11,18,26,35,49,70,89,123,164,212,278,366,460,597,762,
%T A364059 957,1210,1530,1891,2369,2943,3621,4468,5507,6703,8210,10004,12115,
%U A364059 14688,17782,21365,25743,30913,36965,44210,52801,62753,74667,88626,104874,124070
%N A364059 Number of integer partitions of n whose rounded mean is > 1. Partitions with mean >= 3/2.
%C A364059 We use the "rounding half to even" rule, see link.
%H A364059 Wikipedia, <a href="https://en.wikipedia.org/wiki/Rounding">Rounding</a>.
%F A364059 a(n) = A000041(n) - A363947(n).
%e A364059 The a(0) = 0 through a(8) = 18 partitions:
%e A364059   .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)
%e A364059              (21)  (22)  (32)   (33)    (43)    (44)
%e A364059                    (31)  (41)   (42)    (52)    (53)
%e A364059                          (221)  (51)    (61)    (62)
%e A364059                          (311)  (222)   (322)   (71)
%e A364059                                 (321)   (331)   (332)
%e A364059                                 (411)   (421)   (422)
%e A364059                                 (2211)  (511)   (431)
%e A364059                                 (3111)  (2221)  (521)
%e A364059                                         (3211)  (611)
%e A364059                                         (4111)  (2222)
%e A364059                                                 (3221)
%e A364059                                                 (3311)
%e A364059                                                 (4211)
%e A364059                                                 (5111)
%e A364059                                                 (22211)
%e A364059                                                 (32111)
%e A364059                                                 (41111)
%t A364059 Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]>1&]],{n,0,30}]
%Y A364059 Rounding-up gives A000065.
%Y A364059 Rounding-down gives A110618, ranks A344291.
%Y A364059 For median instead of mean we appear to have A238495.
%Y A364059 The complement is counted by A363947, ranks A363948.
%Y A364059 A000041 counts integer partitions.
%Y A364059 A008284 counts partitions by length, A058398 by mean.
%Y A364059 A025065 counts partitions with low mean 1, ranks A363949.
%Y A364059 A067538 counts partitions with integer mean, ranks A316413.
%Y A364059 A124943 counts partitions by low median, high A124944.
%Y A364059 Cf. A002865, A098859, A241131, A327482, A363723, A363724, A363731, A363946.
%K A364059 nonn
%O A364059 0,4
%A A364059 _Gus Wiseman_, Jul 06 2023