%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