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

0,9

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.

Formula

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

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 *)

Crossrefs

Cf. A000035, A000700, A357456.

Sequence in context: A112219 A035458 A259771 * A194902 A194874 A194835

Adjacent sequences: A357454 A357455 A357456 * A357458 A357459 A357460

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 29 2022

Status

approved