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

Author

Brian Hopkins, May 16 2017

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

Cf. A126796.

Programs

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 *)

Formula

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