A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.

0, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 30, 35, 44, 52, 63, 74, 90, 105, 126, 148, 175, 204, 242, 280, 330, 382, 446, 515, 600, 690, 800, 919, 1060, 1214, 1398, 1595, 1830, 2086, 2384, 2711, 3092, 3506, 3988, 4516, 5122, 5788, 6552, 7388, 8345

Offset: 0

Author

Gus Wiseman, Feb 25 2022

Keywords

• nonn

Examples

			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
		

Crossrefs

The ordered version (compositions) is A000213 shifted right once.

All odd-length partitions are counted by A027193.

The opposite appears to be A122130, even-length A351008, any length A122129.

This appears to be the odd-length case of A122135, even-length A122134.

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

Cf. A000041, A000070, A003242, A027383, A236559, A236914, A350842.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]