A340385 - OEIS

A340385

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

%I #6 Jan 09 2021 08:37:32

%S 1,0,2,0,3,1,6,3,10,7,18,15,30,28,51,50,82,87,134,145,211,235,331,375,

%T 510,586,779,901,1172,1366,1750,2045,2581,3026,3778,4433,5476,6430,

%U 7878,9246,11240,13189,15931,18670,22417,26242,31349,36646,43567,50854

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

%e The a(3) = 2 through a(10) = 7 partitions:

%e 3 5 321 7 332 9 532

%e 111 311 322 521 333 541

%e 11111 331 32111 522 721

%e 511 531 32221

%e 31111 711 33211

%e 1111111 32211 52111

%e 33111 3211111

%e 51111

%e 3111111

%e 111111111

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

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

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

%Y The Heinz numbers of these partitions are given by A340386.

%Y Other cases of odd length:

%Y - A024429 counts set partitions of odd length.

%Y - A067659 counts strict partitions of odd length.

%Y - A089677 counts ordered set partitions of odd length.

%Y - A166444 counts compositions of odd length.

%Y - A174726 counts ordered factorizations of odd length.

%Y - A332304 counts strict compositions of odd length.

%Y - A339890 counts factorizations of odd length.

%Y A000009 counts partitions into odd parts, ranked by A066208.

%Y A026804 counts partitions whose least part is odd.

%Y A058695 counts partitions of odd numbers, ranked by A300063.

%Y A072233 counts partitions by sum and length.

%Y A101707 counts partitions with odd rank.

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

%Y A340101 counts factorizations into odd factors.

%Y A340102 counts odd-length factorizations into odd factors.

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Jan 08 2021