A361394 Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).

1, 1, 2, 2, 4, 6, 8, 11, 15, 20, 30, 38, 49, 65, 83, 108, 139, 178, 224, 286, 358, 437, 550, 684, 837, 1037, 1269, 1553, 1889, 2295, 2770, 3359, 4035, 4843, 5808, 6951, 8312, 9902, 11752, 13958, 16531, 19541, 23037, 27162, 31911, 37488, 43950, 51463, 60127, 70229

Offset: 0

Author

Gus Wiseman, Mar 17 2023

Keywords

• nonn

Examples

			The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)   (4)    (5)     (6)     (7)
       (11)  (21)  (22)   (32)    (33)    (43)
                   (31)   (41)    (42)    (52)
                   (211)  (221)   (51)    (61)
                          (311)   (321)   (322)
                          (2111)  (411)   (331)
                                  (2211)  (421)
                                  (3111)  (511)
                                          (2221)
                                          (3211)
                                          (4111)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

The complement is counted by A360254, ranks A360558.

These partitions have ranks A361395.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, reverse A058398.

A067538 counts partitions with integer mean, strict A102627.

A116608 counts partitions by number of distinct parts.

Cf. A106529, A144300, A237363, A237832, A240219, A316413, A349156.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),
     `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 19 2023

Mathematica

Table[Length[Select[IntegerPartitions[n],2*Length[Union[#]]>=Length[#]&]],{n,0,30}]