A391225 Number of integer partitions of n such that the least and greatest parts are both even or both odd.

1, 2, 2, 4, 3, 8, 6, 14, 13, 25, 24, 45, 45, 74, 82, 125, 139, 204, 233, 329, 382, 517, 609, 811, 957, 1242, 1478, 1892, 2249, 2839, 3381, 4221, 5024, 6205, 7386, 9055, 10751, 13078, 15513, 18760, 22198, 26687, 31521, 37710, 44440, 52913, 62228, 73804, 86590

Offset

1,2

Links

Table of n, a(n) for n=1..49.

Example

The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)        (333)
                    (31)    (11111)  (42)      (511)      (53)        (432)
                    (1111)           (51)      (3211)     (62)        (531)
                                     (222)     (31111)    (71)        (711)
                                     (321)     (1111111)  (422)       (3321)
                                     (3111)               (521)       (5211)
                                     (111111)             (2222)      (32211)
                                                          (3221)      (33111)
                                                          (3311)      (51111)
                                                          (5111)      (321111)
                                                          (32111)     (3111111)
                                                          (311111)    (111111111)
                                                          (11111111)

Mathematica

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

Crossrefs

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

For even length we have A027187, ranks A028260, strict A067661.

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

For even sum of prime indices we have A300061, counted by A058696.

For both parts even we have A325346, ranks A391232, strict A391233.

The complement is counted by A390092, ranks A390988.

These partitions are ranked by A391226.

The strict case is A391227, complement counted by A390746.

For product instead of sum we have A391229, counted by A391230, strict A391231.

A026424 ranks partitions of odd length, counted by A027193, strict A067659.

A244991 ranks partitions of odd maximum, counted by A027193, strict A026837.

A390093 ranks partitions of odd min and max, counted by A325338, strict A391228.

A340932 ranks partitions of odd minimum, counted by A026804, strict A026832.

A066207 lists products of even-indexed primes, counted by A035363, strict A035457.

A066208 lists products of odd-indexed primes, counted by A000009, strict A000700.

A390430 gives least prime index plus greatest prime index, for differences A243055.

Cf. A005843, A101707, A160786, A257992, A300272, A340832.

Sequence in context: A056429 A133806 A360241 * A283717 A185333 A005176

Adjacent sequences: A391222 A391223 A391224 * A391226 A391227 A391228

Keyword

nonn

Author

Gus Wiseman, Dec 07 2025

Status

approved