A325724 Number of separable partitions of n that consist of an odd number of parts.

1, 1, 1, 1, 1, 2, 3, 6, 10, 14, 22, 30, 40, 50, 70, 82, 107, 127, 166, 203, 253, 313, 402, 495, 633, 788, 984, 1251, 1551, 1929, 2388, 2963, 3603, 4465, 5412, 6570, 7936, 9617, 11477, 13852, 16482, 19708, 23384, 27830, 32923, 39066, 46078, 54416, 64114

Offset

1,6

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..49.

Formula

a(n) = A325534(n) - A325723(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, A325723.

Sequence in context: A084396 A090035 A337611 * A373271 A111467 A215891

Adjacent sequences: A325721 A325722 A325723 * A325725 A325726 A325727

Keyword

nonn,easy

Author

Clark Kimberling, May 19 2019

Status

approved