A364055 Number of integer partitions of n satisfying (length) = (mean). Partitions of n into sqrt(n) parts.

1, 1, 0, 0, 2, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 0, 0, 0, 0, 192, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1206, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8033, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55974, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Jul 07 2023

Keywords

• nonn

Examples

			The a(0) = 1 through a(9) = 7 partitions:
  ()  (1)  .  .  (22)  .  .  .  .  (333)
                 (31)              (432)
                                   (441)
                                   (522)
                                   (531)
                                   (621)
                                   (711)
		

Crossrefs

The strict case is A107379(sqrt(n)).

Without zeros we have A206240.

These partitions have ranks A363951.

A008284 counts partitions by length, A058398 by mean.

A067538 counts partitions with integer mean, ranks A316413.

Cf. A025065, A026905, A237984, A327472, A327482, A363944, A363949.

Programs

Mathematica

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

Formula

a(n^2) = A206240(n).