A239483 Number of (3,0)-separable partitions of n; see Comments.

0, 1, 1, 1, 1, 3, 3, 4, 5, 7, 8, 10, 12, 16, 18, 22, 26, 33, 38, 45, 53, 65, 75, 89, 103, 124, 143, 168, 195, 230, 265, 309, 357, 418, 479, 556, 639, 742, 850, 979, 1122, 1294, 1478, 1696, 1935, 2220, 2528, 2889, 3287, 3752, 4261, 4850, 5502, 6257, 7084

Offset

4,6

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=4..58.

Example

The (3,0)-separable partitions of 11 are 731, 632, 434, 23231, so that a(11) = 4.

Mathematica

z = 65; -1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 1] == Length[p] - 1], {n, 2, z}]  (* A165652 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 2] == Length[p] - 1], {n, 3, z}]  (* A239482 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 3] == Length[p] - 1], {n, 4, z}]  (* A239483 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 4] == Length[p] - 1], {n, 5, z}]  (* A239484 *)
-1 + Table[Count[IntegerPartitions[n], p_ /; 2 Count[p, 5] == Length[p] - 1], {n, 6, z}] (* A239485 *)

Crossrefs

Cf. A239467, A165652, A239482, A239484, A239485.

Sequence in context: A144489 A239640 A100091 * A104806 A241442 A363091

Adjacent sequences: A239480 A239481 A239482 * A239484 A239485 A239486

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2014

Status

approved