A357457 Number of partitions of n into two or more distinct odd parts.

0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 3, 3, 5, 4, 5, 5, 7, 7, 8, 8, 11, 11, 12, 13, 16, 16, 18, 19, 23, 24, 26, 28, 33, 34, 37, 40, 46, 48, 52, 56, 63, 67, 72, 77, 87, 92, 98, 106, 117, 124, 133, 143, 157, 167, 178, 191, 209, 222, 236, 254, 276, 293, 312, 334

Offset: 0

Author

Ilya Gutkovskiy, Sep 29 2022

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Harmandeep Kaur and Muhammad Asif Rana, Partitions with unique largest part and their generating functions, arXiv:2506.11447 [math.CO], 2025. See p. 6.

Crossrefs

Cf. A000035, A000700, A357456.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(n>i^2, 0,
      b(n, i-1)+`if`(i*2-1>n, 0, b(n-(i*2-1), i-1))))
    end:
a:= n-> b(n, iquo(n+1, 2))-irem(max(1, n), 2):
seq(a(n), n=0..67);  # Alois P. Heinz, Jul 25 2025

Mathematica

Monitor[Table[Count[IntegerPartitions[n], ?(And[Length[#] > 1, DuplicateFreeQ[#], AllTrue[#, OddQ]] &)], {n, 0, 67}], n] (* _Michael De Vlieger, Jun 23 2025 *)

Formula

a(n) = A000700(n) - A000035(n) for n > 0.