A351012 Number of even-length integer partitions y of n such that y_i = y_{i+1} for all even i.

1, 0, 1, 1, 3, 3, 5, 6, 9, 10, 13, 16, 21, 24, 29, 35, 43, 50, 60, 70, 83, 97, 113, 132, 156, 178, 206, 239, 275, 316, 365, 416, 477, 545, 620, 706, 806, 912, 1034, 1173, 1326, 1496, 1691, 1902, 2141, 2410, 2704, 3034, 3406, 3808, 4261, 4765, 5317, 5932, 6617

Offset: 0

Author

Gus Wiseman, Feb 03 2022

Keywords

• nonn

Examples

			The a(2) = 1 through a(8) = 9 partitions:
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (3111)    (2221)    (71)
                              (111111)  (4111)    (2222)
                                        (211111)  (3221)
                                                  (5111)
                                                  (311111)
                                                  (11111111)
		

Crossrefs

The ordered version (compositions) is A027383(n-2).

For odd instead of even indices we have A035363, any length A351004.

The version for unequal parts appears to be A122134, any length A122135.

This is the even-length case of A351003.

Requiring inequalities at odd positions gives A351007, any length A351006.

Cf. A000070, A018819, A035457, A088218, A101417, A122129, A350837, A351005, A351008.

Programs

Mathematica

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