A360670 Number of integer partitions of n whose parts have the same mean as their negated first differences.

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 2, 0, 2, 2, 0, 0, 5, 1, 0, 3, 3, 0, 4, 0, 5, 3, 0, 2, 10, 0, 0, 4, 10, 0, 5, 0, 7, 9, 0, 0, 17, 1, 6, 5, 10, 0, 9, 8, 14, 6, 0, 0, 34, 0, 0, 9, 18, 13, 13, 0, 17, 7, 19, 0, 40, 0, 0, 28

Offset: 0

Author

Gus Wiseman, Feb 18 2023

Keywords

• nonn

Examples

			The a(n) partitions for n = 4, 12, 24, 27, 30, 44:
  (3,1)  (9,3)    (18,6)      (19,7,1)  (21,8,1)      (33,11)
         (9,2,1)  (17,6,1)    (20,5,2)  (22,6,2)      (34,5,4,1)
                  (18,4,2)    (21,3,3)  (23,4,3)      (34,6,3,1)
                  (19,2,2,1)            (25,2,1,1,1)  (34,7,2,1)
                  (19,3,1,1)                          (34,8,1,1)
                                                      (35,4,3,2)
                                                      (35,5,2,2)
For example, the partition y = (28,4,3,1), with mean 9, has negated first differences (24,2,1), with mean 9, so y is counted under a(36).
		

Crossrefs

These partitions have ranks A360669.

A000041 counts integer partitions, strict A000009.

A058398 counts partitions by mean, see also A008284, A327482.

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

A326567/A326568 gives mean of prime indices.

A360614/A360615 = mean of first differences of 0-prepended prime indices.

Cf. A060016, A240219, A116608, A316313, A340610, A360068.

Programs

Mathematica

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