A238495 Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

1, 2, 3, 4, 7, 9, 14, 19, 27, 36, 51, 66, 90, 118, 156, 201, 264, 336, 434, 550, 700, 880, 1112, 1385, 1733, 2149, 2666, 3283, 4049, 4956, 6072, 7398, 9009, 10922, 13237, 15970, 19261, 23147, 27790, 33260, 39776, 47425, 56497, 67133, 79685, 94371, 111653

Offset: 1

Author

Clark Kimberling, Feb 27 2014

Keywords

• nonn
• easy

Comments

Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - Gus Wiseman, Jul 10 2023

Examples

			a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.
From _Gus Wiseman_, Jul 10 2023 (Start)
The a(2) = 1 through a(8) = 14 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)
       (21)  (22)  (32)   (33)    (43)    (44)
             (31)  (41)   (42)    (52)    (53)
                   (221)  (51)    (61)    (62)
                          (222)   (322)   (71)
                          (321)   (331)   (332)
                          (2211)  (421)   (422)
                                  (2221)  (431)
                                  (3211)  (521)
                                          (2222)
                                          (3221)
                                          (3311)
                                          (4211)
                                          (22211)
(End)
		

Crossrefs

Cf. A096373.

For mean instead of median we have A000065, ranks A057716.

The complement is counted by A027336, ranks A364056.

Rows sums of A359893 if we remove the first column.

These partitions have ranks A364058.

A000041 counts integer partitions.

A008284 counts partitions by length, A058398 by mean.

A025065 counts partitions with low mean 1, ranks A363949.

A124943 counts partitions by low median, high A124944.

A241131 counts partitions with low mode 1, ranks A360015.

Cf. A000070, A000975, A002865, A110618, A237984, A363488.

Programs

Mathematica

Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]
Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* Gus Wiseman, Jul 10 2023 *)

Formula

From Gus Wiseman, Jul 11 2023: (Start)

a(n>2) = A000041(n) - A096373(n-2).

a(n>1) = A000041(n-2) + A002865(n+1).

a(n) = A000041(n+1) - A027336(n).

(End)

Extensions

Formula corrected by Gus Wiseman, Jul 11 2023