A391230 Number of integer partitions of n > 0 such that the least and greatest parts are not both odd (equivalently, their product is even).

0, 1, 1, 3, 4, 6, 9, 13, 18, 24, 34, 44, 61, 78, 104, 133, 176, 221, 288, 360, 462, 575, 729, 901, 1132, 1393, 1733, 2120, 2620, 3190, 3913, 4744, 5784, 6983, 8464, 10176, 12276, 14704, 17652, 21064, 25184, 29951, 35664, 42276, 50157, 59274, 70081, 82572

Offset

1,4

Links

Table of n, a(n) for n=1..48.

Example

The a(2) = 1 through a(8) = 13 partitions:
  (2)  (21)  (4)    (32)    (6)      (43)      (8)
             (22)   (41)    (42)     (52)      (44)
             (211)  (221)   (222)    (61)      (62)
                    (2111)  (411)    (322)     (332)
                            (2211)   (421)     (422)
                            (21111)  (2221)    (431)
                                     (4111)    (611)
                                     (22111)   (2222)
                                     (211111)  (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)

Mathematica

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

Crossrefs

For odd least part we have A026804 (strict A026832), ranks A340932.

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

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

For odd greatest part we have A027193 (strict A026837), ranks A244991.

The complement is counted by A325338 (strict A391228), ranks A390093.

For sum instead of product we have A391225 (strict A391227), ranks A391226.

These partitions have ranks A391229.

The strict case is A391231.

A000041 counts integer partitions, strict A000009.

A257991 and A257992 count odd/even prime indices.

A333352 and A390430 give least prime index times/plus greatest prime index.

Cf. A005408, A006141, A035457, A058695, A067659, A101707, A160786, A325346, A390092, A390746, A390988.

Sequence in context: A167928 A090867 A152950 * A005626 A227561 A352095

Adjacent sequences: A391227 A391228 A391229 * A391231 A391232 A391233

Keyword

nonn

Author

Gus Wiseman, Dec 12 2025

Status

approved