A361392 Number of integer partitions of n whose first differences have mean -1.

0, 0, 0, 1, 0, 2, 1, 3, 2, 5, 4, 8, 7, 12, 12, 19, 19, 29, 31, 43, 48, 65, 73, 97, 110, 142, 164, 208, 240, 301, 350, 432, 504, 617, 719, 874, 1019, 1228, 1434, 1717, 2001, 2385, 2778, 3292, 3831, 4522, 5252, 6177, 7164, 8392, 9722, 11352, 13125, 15283, 17643

Offset

0,6

Comments

These are partitions where the first part minus the last part is the number of parts minus 1.

Links

Table of n, a(n) for n=0..54.

Example

The a(3) = 1 through a(11) = 8 partitions:
  (21)  .  (32)   (321)  (43)    (422)   (54)     (442)    (65)
           (311)         (331)   (4211)  (432)    (4321)   (533)
                         (4111)          (4221)   (4411)   (4331)
                                         (4311)   (52111)  (4421)
                                         (51111)           (5222)
                                                           (52211)
                                                           (53111)
                                                           (611111)
For example, the partition y = (4,2,2,1) has first differences (-2,0,-1), with mean -1, so y is counted under a(9).

Mathematica

Table[Length[Select[IntegerPartitions[n], Mean[Differences[#]]==-1&]], {n, 0, 30}]

Crossrefs

For mean 0 we have A032741.

The 0-appended version is A047993.

For any negative mean we have A144300.

A000041 counts integer partitions, strict A000009.

A008284/A058398/A327482 count partitions by mean.

A067538 counts partitions with integer mean, ranks A316413.

A326567/A326568 gives mean of prime indices, conjugate A326839/A326840.

A360614/A360615 gives mean of 0-appended first differences of prime indices.

Cf. A102627, A237363, A237832, A348551, A360688.

Sequence in context: A097451 A005916 A034392 * A181531 A034393 A068932

Adjacent sequences: A361389 A361390 A361391 * A361393 A361394 A361395

Keyword

nonn

Author

Gus Wiseman, Mar 13 2023

Status

approved