login
Number of partitions of n such that each part is no more than 4 more than the sum of all smaller parts.
2

%I #30 May 24 2018 09:30:38

%S 1,1,2,3,5,6,10,13,20,26,37,48,68,86,119,152,204,258,342,428,560,698,

%T 897,1114,1421,1748,2210,2712,3390,4140,5140,6240,7702,9314,11402,

%U 13741,16742,20071,24333,29087,35056,41770,50137,59503,71148,84195,100213,118275,140307,165041,195139

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

%C 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.

%H Alois P. Heinz, <a href="/A286097/b286097.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). - _Vaclav Kotesovec_, May 24 2018

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

%t 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 *)

%Y Cf. A126796.

%K nonn

%O 0,3

%A _Brian Hopkins_, May 16 2017