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

0, 0, 1, 2, 3, 5, 9, 11, 18, 26, 35, 49, 70, 89, 123, 164, 212, 278, 366, 460, 597, 762, 957, 1210, 1530, 1891, 2369, 2943, 3621, 4468, 5507, 6703, 8210, 10004, 12115, 14688, 17782, 21365, 25743, 30913, 36965, 44210, 52801, 62753, 74667, 88626, 104874, 124070

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..47.

Wikipedia, Rounding.

Formula

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

Example

The a(0) = 0 through a(8) = 18 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)     (7)     (8)
             (21)  (22)  (32)   (33)    (43)    (44)
                   (31)  (41)   (42)    (52)    (53)
                         (221)  (51)    (61)    (62)
                         (311)  (222)   (322)   (71)
                                (321)   (331)   (332)
                                (411)   (421)   (422)
                                (2211)  (511)   (431)
                                (3111)  (2221)  (521)
                                        (3211)  (611)
                                        (4111)  (2222)
                                                (3221)
                                                (3311)
                                                (4211)
                                                (5111)
                                                (22211)
                                                (32111)
                                                (41111)

Mathematica

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

Crossrefs

Rounding-up gives A000065.

Rounding-down gives A110618, ranks A344291.

For median instead of mean we appear to have A238495.

The complement is counted by A363947, ranks A363948.

A000041 counts integer partitions.

A008284 counts partitions by length, A058398 by mean.

A025065 counts partitions with low mean 1, ranks A363949.

A067538 counts partitions with integer mean, ranks A316413.

A124943 counts partitions by low median, high A124944.

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

Sequence in context: A329357 A101737 A176550 * A293036 A214125 A275466

Adjacent sequences: A364056 A364057 A364058 * A364060 A364061 A364062

Keyword

nonn

Author

Gus Wiseman, Jul 06 2023

Status

approved