A239495 Number of (4,1)-separable partitions of n; see Comments.

0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 2, 3, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 16, 20, 23, 27, 32, 37, 43, 52, 59, 69, 80, 93, 108, 126, 144, 166, 192, 222, 254, 294, 334, 384, 441, 504, 575, 658, 748, 854, 972, 1106, 1254, 1428, 1617, 1834, 2077, 2350, 2656, 3007

Offset

1,10

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..60.

Example

The (4,1)-separable partitions of 12 are 84, 3414, 2424, so that a(12) = 3.

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, A239494, A239496.

Sequence in context: A335967 A319611 A337102 * A348163 A298210 A045772

Adjacent sequences: A239492 A239493 A239494 * A239496 A239497 A239498

Keyword

nonn,easy

Author

Clark Kimberling, Mar 20 2014

Status

approved