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

0,13

Links

Table of n, a(n) for n=0..75.

Example

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).

Mathematica

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

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.

Sequence in context: A078646 A264405 A304650 * A347992 A317581 A359865

Adjacent sequences: A360667 A360668 A360669 * A360671 A360672 A360673

Keyword

nonn

Author

Gus Wiseman, Feb 18 2023

Status

approved