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Number of integer partitions of n where 2*(number of distinct parts) >= (number of parts).
15

%I #10 Mar 19 2023 18:09:06

%S 1,1,2,2,4,6,8,11,15,20,30,38,49,65,83,108,139,178,224,286,358,437,

%T 550,684,837,1037,1269,1553,1889,2295,2770,3359,4035,4843,5808,6951,

%U 8312,9902,11752,13958,16531,19541,23037,27162,31911,37488,43950,51463,60127,70229

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

%H Alois P. Heinz, <a href="/A361394/b361394.txt">Table of n, a(n) for n = 0..1000</a>

%e The a(1) = 1 through a(7) = 11 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (21) (22) (32) (33) (43)

%e (31) (41) (42) (52)

%e (211) (221) (51) (61)

%e (311) (321) (322)

%e (2111) (411) (331)

%e (2211) (421)

%e (3111) (511)

%e (2221)

%e (3211)

%e (4111)

%p b:= proc(n, i, t) option remember; `if`(n=0, `if`(t>=0, 1, 0),

%p `if`(i<1, 0, add(b(n-i*j, i-1, t+`if`(j>0, 2, 0)-j), j=0..n/i)))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Mar 19 2023

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

%Y The complement is counted by A360254, ranks A360558.

%Y These partitions have ranks A361395.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, reverse A058398.

%Y A067538 counts partitions with integer mean, strict A102627.

%Y A116608 counts partitions by number of distinct parts.

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Mar 17 2023