A239494 Number of (3,1)-separable partitions of n; see Comments.

0, 0, 0, 1, 1, 0, 1, 2, 2, 2, 2, 4, 4, 5, 6, 8, 9, 11, 13, 17, 19, 23, 27, 34, 39, 46, 54, 66, 76, 90, 104, 125, 144, 169, 196, 231, 266, 310, 358, 419, 480, 557, 640, 743, 851, 980, 1123, 1295, 1479, 1697, 1936, 2221, 2529, 2890, 3288, 3753, 4262, 4851

Offset

1,8

Comments

Suppose that p is a partition of n into 2 or more parts and that h is a part of p. Then p is (h,0)-separable if there is an ordering x, h, x, h, ..., h, x of the parts of p, where each x represents any part of p except h. Here, the number of h's on the ends of the ordering is 0. Similarly, p is (h,1)-separable if there is an ordering x, h, x, h, ..., x, h, where the number of h's on the ends is 1; next, p is (h,2)-separable if there is an ordering h, x, h, ..., x, h. Finally, p is h-separable if it is (h,i)-separable for i = 0,1,2.

Links

Table of n, a(n) for n=1..58.

Example

The (3,1)-separable partitions of 14 are [11,3], [7,3,1,3], [6,3,2,3], [4,3,4,3], [2,3,2,3,1,3], so that a(14) = 5.

Mathematica

z = 70; Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p]], {n, 1, z}] (* A008483 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p]], {n, 1, z}] (* A239493 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p]], {n, 1, z}] (* A239494 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p]], {n, 1, z}] (* A239495 *)
Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p]], {n, 1, z}] (* A239496 *)

Crossrefs

Cf. A230467, A008483, A239493, A239495, A239496.

Sequence in context: A355806 A032576 A276420 * A349388 A347663 A071809

Adjacent sequences: A239491 A239492 A239493 * A239495 A239496 A239497

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2014

Status

approved