A351007 Number of even-length integer partitions of n into parts that are alternately unequal and equal.

1, 0, 0, 1, 1, 2, 2, 3, 4, 5, 5, 7, 8, 9, 10, 13, 14, 16, 18, 20, 23, 27, 28, 32, 37, 40, 44, 51, 54, 60, 67, 73, 81, 90, 96, 107, 118, 127, 139, 154, 166, 181, 198, 213, 232, 256, 273, 297, 325, 348, 377, 411, 440, 476, 516, 555, 598, 647, 692, 746, 807

Offset

0,6

Comments

These are partitions whose multiplicities begin with a 1, are followed by any number of 2's, and end with another 1.

Links

Table of n, a(n) for n=0..60.

Example

The a(3) = 1 through a(15) = 13 partitions (A..E = 10..14):
  21  31  32  42  43  53    54    64    65    75    76    86    87
          41  51  52  62    63    73    74    84    85    95    96
                  61  71    72    82    83    93    94    A4    A5
                      3221  81    91    92    A2    A3    B3    B4
                            4221  5221  A1    B1    B2    C2    C3
                                        4331  4332  C1    D1    D2
                                        6221  5331  5332  5441  E1
                                              7221  6331  6332  5442
                                                    8221  7331  6441
                                                          9221  7332
                                                                8331
                                                                A221
                                                                433221

Mathematica

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

Crossrefs

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

This is the even-length case of A351006, odd-length A053251.

Without equalities we have A351008, any length A122129, opposite A122135.

Without inequalities we have A351012, any length A351003, opposite A351004.

Cf. A000070, A003242, A018819, A027383, A035363, A088218, A122134, A350842, A350844, A351011.

Sequence in context: A112341 A242774 A183003 * A307779 A165684 A342519

Adjacent sequences: A351004 A351005 A351006 * A351008 A351009 A351010

Keyword

nonn

Author

Gus Wiseman, Jan 31 2022

Status

approved