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

Author

Gus Wiseman, Aug 01 2021

Keywords

• nonn

Examples

			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)
		

Links

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

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.

Programs

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}]

Extensions

More terms from Alois P. Heinz, Aug 05 2021