A286097 Number of partitions of n such that each part is no more than 4 more than the sum of all smaller parts.

1, 1, 2, 3, 5, 6, 10, 13, 20, 26, 37, 48, 68, 86, 119, 152, 204, 258, 342, 428, 560, 698, 897, 1114, 1421, 1748, 2210, 2712, 3390, 4140, 5140, 6240, 7702, 9314, 11402, 13741, 16742, 20071, 24333, 29087, 35056, 41770, 50137, 59503, 71148, 84195, 100213, 118275, 140307, 165041, 195139

Offset

0,3

Comments

Generalization of Adams-Watters's criterion for complete partitions, that each part is no more than 1 more than the sum of all smaller parts.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, May 24 2018

Example

For n = 8, a(8) = 20 counts all partitions of 8 except (8) and (7,1).

Mathematica

Table[Count[IntegerPartitions@n, w_ /; And[Last@w <= 4, NoneTrue[ w - Rest@  PadRight[4 + Reverse@Accumulate@Reverse@w, Length@w + 1, Last@w], # > 0 &]]], {n, 50}] (* George Beck, May 17 2017, Version 11.1.1, adapted from A286929 *)

Crossrefs

Cf. A126796.

Sequence in context: A145724 A039843 A305937 * A287483 A014853 A131627

Adjacent sequences: A286094 A286095 A286096 * A286098 A286099 A286100

Keyword

nonn

Author

Brian Hopkins, May 16 2017

Status

approved