A325723 Number of separable partitions of n that consist of an even number of parts.

0, 0, 1, 2, 4, 4, 7, 8, 9, 12, 15, 19, 26, 37, 46, 70, 91, 127, 163, 219, 283, 365, 456, 582, 716, 893, 1105, 1336, 1642, 1998, 2432, 2934, 3588, 4284, 5211, 6291, 7599, 9107, 11041, 13177, 15891, 18989, 22790, 27168, 32459, 38535, 45872, 54361, 64362, 76021

Offset

1,4

Comments

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

Links

Table of n, a(n) for n=1..50.

Formula

a(n) = A325534(n) - A325724(n) for n >= 1.

Mathematica

(separable = Table[Map[#[[1]] &, Select[Map[{#,
Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]]} &,
IntegerPartitions[nn]], #[[2]] > 1 &]], {nn, 60}]) //  ColumnForm;
Map[Map[#[[1]] &, Select[Map[{#, Length[Union[#]]} &, #], #[[2]] == 2 &]] &,
separable] // ColumnForm;
Map[Length[Select[Map[{#, Length[Union[#]]} &, #], EvenQ[#[[2]]] &]] &,
  separable] (* A325723 *)
Map[Length[Select[Map[{#, Length[Union[#]]} &, #], OddQ[#[[2]]] &]] &,
  separable]  (* A325724 *)
(* Peter J. C. Moses, May 08 2019 *)

Crossrefs

Cf. A000041, A325534, A325724.

Sequence in context: A266188 A272881 A325646 * A353927 A262884 A340448

Adjacent sequences: A325720 A325721 A325722 * A325724 A325725 A325726

Keyword

nonn,easy

Author

Clark Kimberling, May 19 2019

Status

approved