A360683 Number of integer partitions of n whose second differences sum to 0, meaning either there is only one part, or the first two parts have the same difference as the last two parts.

1, 1, 2, 3, 4, 4, 8, 6, 11, 12, 17, 14, 32, 23, 40, 44, 64, 59, 104, 93, 149, 157, 218, 227, 342, 349, 481, 538, 713, 777, 1052, 1145, 1494, 1692, 2130, 2416, 3064, 3449, 4286, 4918, 6028, 6882, 8424, 9620, 11634, 13396, 16022, 18416, 22019, 25248, 29954

Offset

0,3

Links

Table of n, a(n) for n=0..50.

Example

The a(1) = 1 through a(8) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (11111)  (51)      (61)       (62)
                                     (222)     (22111)    (71)
                                     (321)     (1111111)  (2222)
                                     (2211)               (3221)
                                     (111111)             (3311)
                                                          (22211)
                                                          (221111)
                                                          (11111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Total[Differences[#, 2]]==0&]], {n, 0, 30}]

Crossrefs

For mean instead of sum we have a(n) - A008619(n).

For median instead of sum we have A360682.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by number of parts.

A067538 counts partitions with integer mean, strict A102627.

A316413 ranks partitions with integer mean, complement A348551.

Cf. A114638, A240219, A325347, A360068.

Sequence in context: A028298 A047966 A360243 * A317085 A236543 A360245

Adjacent sequences: A360680 A360681 A360682 * A360684 A360685 A360686

Keyword

nonn

Author

Gus Wiseman, Feb 19 2023

Status

approved