A325723 - OEIS

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

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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 *)