A364059 - OEIS

%I #12 Jul 08 2023 23:06:19

%S 0,0,1,2,3,5,9,11,18,26,35,49,70,89,123,164,212,278,366,460,597,762,

%T 957,1210,1530,1891,2369,2943,3621,4468,5507,6703,8210,10004,12115,

%U 14688,17782,21365,25743,30913,36965,44210,52801,62753,74667,88626,104874,124070

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

%C We use the "rounding half to even" rule, see link.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Rounding">Rounding</a>.

%F a(n) = A000041(n) - A363947(n).

%e The a(0) = 0 through a(8) = 18 partitions:

%e   .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)

%e              (21)  (22)  (32)   (33)    (43)    (44)

%e                    (31)  (41)   (42)    (52)    (53)

%e                          (221)  (51)    (61)    (62)

%e                          (311)  (222)   (322)   (71)

%e                                 (321)   (331)   (332)

%e                                 (411)   (421)   (422)

%e                                 (2211)  (511)   (431)

%e                                 (3111)  (2221)  (521)

%e                                         (3211)  (611)

%e                                         (4111)  (2222)

%e                                                 (3221)

%e                                                 (3311)

%e                                                 (4211)

%e                                                 (5111)

%e                                                 (22211)

%e                                                 (32111)

%e                                                 (41111)

%t Table[Length[Select[IntegerPartitions[n],Round[Mean[#]]>1&]],{n,0,30}]

%Y Rounding-up gives A000065.

%Y Rounding-down gives A110618, ranks A344291.

%Y For median instead of mean we appear to have A238495.

%Y The complement is counted by A363947, ranks A363948.

%Y A000041 counts integer partitions.

%Y A008284 counts partitions by length, A058398 by mean.

%Y A025065 counts partitions with low mean 1, ranks A363949.

%Y A067538 counts partitions with integer mean, ranks A316413.

%Y A124943 counts partitions by low median, high A124944.

%Y Cf. A002865, A098859, A241131, A327482, A363723, A363724, A363731, A363946.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 06 2023