A365828 Number of strict integer partitions of 2n not containing n.

1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 78, 108, 148, 201, 270, 359, 475, 623, 811, 1050, 1351, 1728, 2201, 2789, 3517, 4418, 5527, 6887, 8553, 10585, 13055, 16055, 19685, 24065, 29343, 35685, 43287, 52387, 63253, 76200, 91605, 109897, 131575, 157231, 187539

Offset

0,3

Links

Table of n, a(n) for n=0..45.

Formula

a(n) = A000009(2n) - A000009(n) + 1.

Example

The a(0) = 1 through a(6) = 12 strict partitions:
  ()  (2)  (4)    (6)    (8)      (10)       (12)
           (3,1)  (4,2)  (5,3)    (6,4)      (7,5)
                  (5,1)  (6,2)    (7,3)      (8,4)
                         (7,1)    (8,2)      (9,3)
                         (5,2,1)  (9,1)      (10,2)
                                  (6,3,1)    (11,1)
                                  (7,2,1)    (5,4,3)
                                  (4,3,2,1)  (7,3,2)
                                             (7,4,1)
                                             (8,3,1)
                                             (9,2,1)
                                             (5,4,2,1)

Mathematica

Table[Length[Select[IntegerPartitions[2n], UnsameQ@@#&&FreeQ[#, n]&]], {n, 0, 30}]

Crossrefs

The complement is counted by A111133.

For non-strict partitions we have A182616, complement A000041.

A000009 counts strict integer partitions.

A046663 counts partitions with no submultiset summing to k, strict A365663.

A365827 counts strict partitions not of length 2, complement A140106.

Cf. A008967, A035363, A078408, A079122, A231429, A238628, A344415, A365659.

Sequence in context: A078408 A007478 A014605 * A232477 A232478 A232476

Adjacent sequences: A365825 A365826 A365827 * A365829 A365830 A365831

Keyword

nonn

Author

Gus Wiseman, Sep 20 2023

Status

approved