OFFSET
0,6
EXAMPLE
The a(1) = 1 through a(12) = 10 partitions (A..C = 10..12):
1 2 3 4 5 6 7 8 9 A B C
221 321 331 332 432 442 443 543
421 431 441 532 542 552
521 531 541 551 642
621 631 632 651
721 641 732
731 741
821 831
33221 921
43221
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]], {i, 2, Length[#]-1, 2}]&]], {n, 0, 30}]
CROSSREFS
The ordered version (compositions) is A000213 shifted right once.
All odd-length partitions are counted by A027193.
The case that is constant at odd indices:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
For equality instead of inequality:
- any length: A351003
- odd-length: A000009 (except at 0)
- even-length: A351012
- opposite any length: A351004
- opposite odd-length: A351594
- opposite even-length: A035363
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 25 2022
STATUS
approved