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Number of separable partitions of n; see Comments.
113

%I #19 Jan 31 2024 22:23:30

%S 1,1,1,2,3,5,6,10,14,19,26,37,49,66,87,116,152,198,254,329,422,536,

%T 678,858,1077,1349,1681,2089,2587,3193,3927,4820,5897,7191,8749,10623,

%U 12861,15535,18724,22518,27029,32373,38697,46174,54998,65382,77601,91950,108777

%N Number of separable partitions of n; see Comments.

%C Definition: a partition is separable if there is an ordering of its parts in which no consecutive parts are identical; otherwise the partition is inseparable.

%C A partition with k parts is separable if and only if there is no part whose multiplicity is greater than ceiling(k/2). - _Andrew Howroyd_, Jan 31 2024

%H Andrew Howroyd, <a href="/A325534/b325534.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) + A325535(n) = A000041(n) = number of partitions of n.

%e For n=5, the partition 1+2+2 is separable as 2+1+2, and 2+1+1+1 is inseparable.

%t Table[Length[Select[Map[Quotient[(1 + Length[#]), Max[Map[Length, Split[#]]]] &,

%t IntegerPartitions[nn]], # > 1 &]], {nn, 50}] (* _Peter J. C. Moses_, May 07 2019 *)

%Y Cf. A000041, A238589, A325535.

%K nonn,easy

%O 0,4

%A _Clark Kimberling_, May 08 2019

%E a(0)=1 prepended by _Alois P. Heinz_, Jan 20 2024