A346634 Number of strict odd-length integer partitions of 2n + 1.

1, 1, 1, 2, 4, 6, 9, 14, 19, 27, 38, 52, 71, 96, 128, 170, 224, 293, 380, 491, 630, 805, 1024, 1295, 1632, 2048, 2560, 3189, 3958, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29250, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Example

The a(0) = 1 through a(7) = 14 partitions:
  (1)  (3)  (5)  (7)      (9)      (11)     (13)      (15)
                 (4,2,1)  (4,3,2)  (5,4,2)  (6,4,3)   (6,5,4)
                          (5,3,1)  (6,3,2)  (6,5,2)   (7,5,3)
                          (6,2,1)  (6,4,1)  (7,4,2)   (7,6,2)
                                   (7,3,1)  (7,5,1)   (8,4,3)
                                   (8,2,1)  (8,3,2)   (8,5,2)
                                            (8,4,1)   (8,6,1)
                                            (9,3,1)   (9,4,2)
                                            (10,2,1)  (9,5,1)
                                                      (10,3,2)
                                                      (10,4,1)
                                                      (11,3,1)
                                                      (12,2,1)
                                                      (5,4,3,2,1)

Maple

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, t, add(b(n-i*j, i-1, abs(t-j)), j=0..min(n/i, 1))))
    end:
a:= n-> b(2*n+1$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 05 2021

Mathematica

Table[Length[Select[IntegerPartitions[2n+1], UnsameQ@@#&&OddQ[Length[#]]&]], {n, 0, 15}]

Crossrefs

Odd bisection of A067659, which is ranked by A030059.

The even version is the even bisection of A067661.

The case of all odd parts is counted by A069911 (non-strict: A078408).

The non-strict version is A160786, ranked by A340931.

The non-strict even version is A236913, ranked by A340784.

The even-length version is A343942 (non-strict: A236914).

The even-sum version is A344650 (non-strict: A236559 or A344611).

A000009 counts partitions with all odd parts, ranked by A066208.

A000009 counts strict partitions, ranked by A005117.

A027193 counts odd-length partitions, ranked by A026424.

A027193 counts odd-maximum partitions, ranked by A244991.

A058695 counts partitions of odd numbers, ranked by A300063.

A340385 counts partitions with odd length and maximum, ranked by A340386.

Other cases of odd length:

- A024429 set partitions

- A089677 ordered set partitions

- A166444 compositions

- A174726 ordered factorizations

- A332304 strict compositions

- A339890 factorizations

Cf. A000700, A008289, A047993, A072233, A106529, A168659, A218171, A340102, A340604, A340607.

Sequence in context: A003402 A328863 A218004 * A034748 A069916 A153140

Adjacent sequences: A346631 A346632 A346633 * A346635 A346636 A346637

Keyword

nonn

Author

Gus Wiseman, Aug 01 2021

Extensions

More terms from Alois P. Heinz, Aug 05 2021

Status

approved