A351008 Alternately strict partitions. Number of even-length integer partitions y of n such that y_i > y_{i+1} for all odd i.

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 19, 23, 28, 34, 41, 50, 60, 71, 85, 102, 120, 142, 168, 197, 231, 271, 316, 369, 429, 497, 577, 668, 770, 888, 1023, 1175, 1348, 1545, 1767, 2020, 2306, 2626, 2990, 3401, 3860, 4379, 4963, 5616, 6350, 7173, 8093

Offset

0,6

Links

Table of n, a(n) for n=0..55.

Formula

Conjecture: a(n+1) = A122129(n+1) - A122130(n). - Gus Wiseman, Feb 21 2022

Example

The a(3) = 1 through a(13) = 12 partitions (A..C = 10..12):
  21   31   32   42   43   53     54     64     65     75     76
            41   51   52   62     63     73     74     84     85
                      61   71     72     82     83     93     94
                           3221   81     91     92     A2     A3
                                  4221   4321   A1     B1     B2
                                         5221   4331   4332   C1
                                                5321   5331   5332
                                                6221   5421   5431
                                                       6321   6331
                                                       7221   6421
                                                              7321
                                                              8221

Mathematica

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

Crossrefs

The version for equal instead of unequal is A035363.

The alternately equal and unequal version is A035457, any length A351005.

This is the even-length case of A122129, opposite A122135.

The odd-length version appears to be A122130.

The alternately unequal and equal version is A351007, any length A351006.

Cf. A000070, A003242, A018819, A027383, A053251, A122134, A350842, A350844, A351003, A351004, A351012.

Sequence in context: A092833 A280664 A100926 * A258875 A179241 A316082

Adjacent sequences: A351005 A351006 A351007 * A351009 A351010 A351011

Keyword

nonn

Author

Gus Wiseman, Jan 31 2022

Status

approved