OFFSET
1,6
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1000
FORMULA
T(n,k) = T(n,n-k) for k=1..n-1, n >= 2.
EXAMPLE
First 13 rows:
1
0 1
1 1 2
1 0 1 2
1 1 1 1 3
2 2 1 2 2 4
2 2 2 2 2 2 5
3 2 3 1 3 2 3 6
3 3 4 3 3 4 3 3 8
5 4 5 4 3 4 5 4 5 10
5 5 5 5 5 5 5 5 5 5 12
7 6 7 7 7 4 7 7 7 6 7 15
8 7 8 8 8 8 8 8 8 8 7 8 18
T(12,4) = 7 counts these partitions: [8,4], [8,3,1], [7,4,1], [6,4,2], [6,3,2,1], [5,4,3], [5,4,2,1].
MATHEMATICA
Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &, Select[IntegerPartitions[nn], # == DeleteDuplicates[#] &]]; Table[Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], sumTo]] >= 1 &]]], {sumTo, nn}], {nn, 45}] // TableForm
u = Flatten[%] (* Peter J. C. Moses, Feb 04 2014 *)
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#], k]&]], {n, 6}, {k, n}] (* Gus Wiseman, Nov 16 2023 *)
CROSSREFS
Column k = n is A000009.
Column k = 2 is A015744.
Column k = 1 is A025147.
The non-strict complement is obtained by adding zeros after A046663.
Diagonal n = 2k is A237258.
Row sums are A284640.
For subsets instead of partitions we have A365381.
The non-strict version is obtained by removing column k = 0 from A365543.
Including column k = 0 gives A365661.
The complement is obtained by adding zeros after A365663.
KEYWORD
AUTHOR
Clark Kimberling, Feb 05 2014
STATUS
approved