A239469 Number of 3-separable partitions of n; see Comments.

0, 0, 0, 1, 2, 1, 3, 4, 5, 6, 8, 11, 13, 15, 20, 24, 30, 35, 43, 52, 63, 74, 89, 106, 127, 148, 177, 208, 246, 287, 338, 396, 464, 538, 630, 732, 853, 985, 1145, 1324, 1532, 1765, 2038, 2345, 2702, 3098, 3562, 4081, 4679, 5348, 6120, 6987, 7978, 9087, 10359

Offset

1,5

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..55.

Example

(3,0)-separable partitions of 7: 232;
(3,1)-separable partitions of 7: 43;
(3,2)-separable partitions of 7: 3231;
3-separable partitions of 7: 232, 43, 3231, so that a(7) = 3.

Mathematica

z = 55; t1 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 1] <= Length[p] + 1], {n, 1, z}] (* A239467 *)
t2 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 2] <= Length[p] + 1], {n, 1, z}] (* A239468 *)
t3 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 3] <= Length[p] + 1], {n, 1, z}] (* A239469 *)
t4 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 4] <= Length[p] + 1], {n, 1, z}] (* A239470 *)
t5 = -1 + Table[Count[IntegerPartitions[n], p_ /; Length[p] - 1 <= 2 Count[p, 5] <= Length[p] + 1], {n, 1, z}] (* A239472 *)

Crossrefs

Cf. A239467, A239468, A239470, A239471.

Sequence in context: A109919 A345903 A346786 * A282840 A370655 A082750

Adjacent sequences: A239466 A239467 A239468 * A239470 A239471 A239472

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2014

Status

approved