A340385 Number of integer partitions of n into an odd number of parts, the greatest of which is odd.

1, 0, 2, 0, 3, 1, 6, 3, 10, 7, 18, 15, 30, 28, 51, 50, 82, 87, 134, 145, 211, 235, 331, 375, 510, 586, 779, 901, 1172, 1366, 1750, 2045, 2581, 3026, 3778, 4433, 5476, 6430, 7878, 9246, 11240, 13189, 15931, 18670, 22417, 26242, 31349, 36646, 43567, 50854

Offset: 1

Author

Gus Wiseman, Jan 08 2021

Keywords

• nonn

Examples

			The a(3) = 2 through a(10) = 7 partitions:
  3     5       321   7         332     9           532
  111   311           322       521     333         541
        11111         331       32111   522         721
                      511               531         32221
                      31111             711         33211
                      1111111           32211       52111
                                        33111       3211111
                                        51111
                                        3111111
                                        111111111
		

Crossrefs

Partitions of odd length are counted by A027193, ranked by A026424.

Partitions with odd maximum are counted by A027193, ranked by A244991.

The Heinz numbers of these partitions are given by A340386.

Other cases of odd length:

- A024429 counts set partitions of odd length.

- A067659 counts strict partitions of odd length.

- A089677 counts ordered set partitions of odd length.

- A166444 counts compositions of odd length.

- A174726 counts ordered factorizations of odd length.

- A332304 counts strict compositions of odd length.

- A339890 counts factorizations of odd length.

A000009 counts partitions into odd parts, ranked by A066208.

A026804 counts partitions whose least part is odd.

A058695 counts partitions of odd numbers, ranked by A300063.

A072233 counts partitions by sum and length.

A101707 counts partitions with odd rank.

A160786 counts odd-length partitions of odd numbers, ranked by A300272.

A340101 counts factorizations into odd factors.

A340102 counts odd-length factorizations into odd factors.

Cf. A000700, A027187, A078408, A174725, A236914.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]*Max[#]]&]],{n,30}]