A351593 Number of odd-length integer partitions of n into parts that are alternately equal and strictly decreasing.

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 2, 4, 3, 5, 4, 6, 4, 8, 6, 9, 6, 12, 7, 14, 10, 16, 11, 20, 13, 24, 16, 28, 18, 34, 21, 40, 26, 46, 30, 56, 34, 64, 41, 75, 48, 88, 54, 102, 64, 118, 73, 138, 84, 159, 98, 182, 112, 210, 128, 242, 148, 276, 168, 318

Offset

0,6

Comments

Also odd-length partitions whose run-lengths are all 2's, except for the last, which is 1.

Links

Table of n, a(n) for n=0..65.

Example

The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3  4  5    6  7    8    9    A    B      C    D      E    F
              221     331  332  441  442  443    552  553    554  663
                                          551         661    662  771
                                          33221       44221       44331
                                                                  55221

Mathematica

Table[Length[Select[IntegerPartitions[n], OddQ[Length[#]]&&And@@Table[If[EvenQ[i], #[[i]]!=#[[i+1]], #[[i]]==#[[i+1]]], {i, Length[#]-1}]&]], {n, 0, 30}]

Crossrefs

The even-length ordered version is A003242, ranked by A351010.

The opposite version is A053251, even-length A351007, any length A351006.

This is the odd-length case of A351005, even-length A035457.

With only equalities we get:

- opposite any length: A351003

- opposite odd-length: A000009 (except at 0)

- opposite even-length: A351012

- any length: A351004

- odd-length: A351594

- even-length: A035363

Without equalities we get:

- opposite any length: A122129 (apparently)

- opposite odd-length: A122130 (apparently)

- opposite even-length: A351008

- any length: A122135 (apparently)

- odd-length: A351595

- even-length: A122134 (apparently)

Cf. A000070, A000984, A027383, A053738, A236559, A236914, A350842, A350844.

Sequence in context: A103858 A010554 A062610 * A025801 A060548 A140426

Adjacent sequences: A351590 A351591 A351592 * A351594 A351595 A351596

Keyword

nonn

Author

Gus Wiseman, Feb 23 2022

Status

approved