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

1,2

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

Links

Table of n, a(n) for n=1..47.

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)

Example

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)

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 *)

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.

Sequence in context: A108950 A238545 A325343 * A239329 A094093 A240077

Adjacent sequences: A238492 A238493 A238494 * A238496 A238497 A238498

Keyword

nonn,easy

Author

Clark Kimberling, Feb 27 2014

Extensions

Formula corrected by Gus Wiseman, Jul 11 2023

Status

approved