A391233 Number of strict integer partitions of n > 0 such that the least part and the greatest part are both even.

0, 1, 0, 1, 0, 2, 0, 2, 1, 3, 1, 4, 2, 5, 4, 7, 6, 9, 8, 13, 12, 16, 17, 23, 24, 29, 32, 40, 43, 52, 57, 69, 76, 88, 100, 117, 129, 148, 167, 193, 215, 245, 274, 313, 349, 394, 442, 500, 557, 625, 698, 784, 873, 977, 1087, 1216, 1349, 1502, 1669, 1858

Offset

1,6

Links

Table of n, a(n) for n=1..60.

Example

The a(10) = 3 through a(16) = 7 strict partitions:
  (10)   (6,3,2)  (12)     (6,5,2)  (14)     (6,5,4)    (16)
  (6,4)           (8,4)    (8,3,2)  (8,6)    (8,5,2)    (10,6)
  (8,2)           (10,2)            (10,4)   (10,3,2)   (12,4)
                  (6,4,2)           (12,2)   (6,4,3,2)  (14,2)
                                    (8,4,2)             (8,6,2)
                                                        (10,4,2)
                                                        (6,5,3,2)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&EvenQ[First[#]]&&EvenQ[Last[#]]&]], {n, 30}]

Crossrefs

For odd least part we have A026832, non-strict A026804, ranks A340932.

For even least part we have A026833, non-strict A026805, ranks A340933.

For odd greatest part we have A026837, non-strict A027193, ranks A244991.

For even greatest part we have A067661, non-strict A027187, ranks A244990.

The non-strict version is A325346, ranks A391232.

For sum instead of product we have A390746, non-strict A390092, ranks A390988.

For both parts odd we have A391228, non-strict A325338, ranks A390093.

A000041 counts integer partitions, strict A000009.

A257991 counts odd prime indices, even A257992.

Cf. A000700, A005843, A035457, A058696, A066207, A067659, A160786, A300061, A390430, A391227, A391231.

Sequence in context: A029221 A304034 A029183 * A213423 A339374 A265753

Adjacent sequences: A391230 A391231 A391232 * A391234 A391235 A391236

Keyword

nonn

Author

Gus Wiseman, Dec 10 2025

Status

approved