A363947 Number of integer partitions of n with mean < 3/2.

0, 1, 1, 1, 2, 2, 2, 4, 4, 4, 7, 7, 7, 12, 12, 12, 19, 19, 19, 30, 30, 30, 45, 45, 45, 67, 67, 67, 97, 97, 97, 139, 139, 139, 195, 195, 195, 272, 272, 272, 373, 373, 373, 508, 508, 508, 684, 684, 684, 915, 915, 915, 1212, 1212, 1212, 1597, 1597, 1597, 2087

Offset

0,5

Links

Table of n, a(n) for n=0..58.

Example

The partition y = (2,2,1) has mean 5/3, which is not less than 3/2, so y is not counted under 5.
The a(1) = 1 through a(8) = 4 partitions:
  (1)  (11)  (111)  (211)   (2111)   (21111)   (22111)    (221111)
                    (1111)  (11111)  (111111)  (31111)    (311111)
                                               (211111)   (2111111)
                                               (1111111)  (11111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Round[Mean[#]]==1&]], {n, 0, 15}]

Crossrefs

The high version is A000012 (all ones).

This is A000070 with each term repeated three times (see A025065 for two).

These partitions have ranks A363948.

The complement is counted by A364059.

A008284 counts partitions by length, A058398 by mean.

A051293 counts subsets with integer mean, median A000975.

A067538 counts partitions with integer mean, strict A102627, ranks A316413.

A327482 counts partitions by integer mean.

A349156 counts partitions with non-integer mean, ranks A348551.

Cf. A000041, A002865, A026905, A027336, A237984, A241131, A327472, A363724, A363745, A363943, A363949.

Sequence in context: A053282 A218084 A240046 * A001584 A180019 A274496

Adjacent sequences: A363944 A363945 A363946 * A363948 A363949 A363950

Keyword

nonn

Author

Gus Wiseman, Jul 02 2023

Status

approved