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

Author

Gus Wiseman, Feb 19 2023

Keywords

• nonn

Examples

			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)
		

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.

Programs

Mathematica

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