This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A365923 #7 Sep 25 2023 12:56:02
%S A365923 1,1,0,1,1,0,2,0,1,0,2,1,1,1,0,4,0,2,0,1,0,5,1,0,3,1,1,0,8,0,3,0,3,0,
%T A365923 1,0,10,2,1,2,2,3,1,1,0,16,0,5,0,3,0,5,0,1,0,20,2,2,4,2,6,0,4,1,1,0,
%U A365923 31,0,6,0,8,0,5,0,5,0,1,0,39,4,4,4,1,6,6,3,2,6,1,1,0
%N A365923 Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums.
%C A365923 For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.
%e A365923 The partition (4,2) has subset-sums {2,4,6} and non-subset-sums {1,3,5} so is counted under T(6,3).
%e A365923 Triangle begins:
%e A365923 1
%e A365923 1 0
%e A365923 1 1 0
%e A365923 2 0 1 0
%e A365923 2 1 1 1 0
%e A365923 4 0 2 0 1 0
%e A365923 5 1 0 3 1 1 0
%e A365923 8 0 3 0 3 0 1 0
%e A365923 10 2 1 2 2 3 1 1 0
%e A365923 16 0 5 0 3 0 5 0 1 0
%e A365923 20 2 2 4 2 6 0 4 1 1 0
%e A365923 31 0 6 0 8 0 5 0 5 0 1 0
%e A365923 39 4 4 4 1 6 6 3 2 6 1 1 0
%e A365923 55 0 13 0 8 0 12 0 6 0 6 0 1 0
%e A365923 71 5 8 7 3 5 3 16 3 6 0 6 1 1 0
%e A365923 Row n = 6 counts the following partitions:
%e A365923 (321) (411) . (51) (33) (6) .
%e A365923 (3111) (42)
%e A365923 (2211) (222)
%e A365923 (21111)
%e A365923 (111111)
%t A365923 Table[Length[Select[IntegerPartitions[n], Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n,0,10}, {k,0,n}]
%Y A365923 Row sums are A000041.
%Y A365923 The rank statistic counted by this triangle is A325799.
%Y A365923 The strict case is A365545, weighted row sums A365922.
%Y A365923 The complement (positive subset-sum) is A365658.
%Y A365923 Weighted row sums are A365918, for positive subset-sums A304792.
%Y A365923 A046663 counts partitions w/o a submultiset summing to k, strict A365663.
%Y A365923 A126796 counts complete partitions, ranks A325781, strict A188431.
%Y A365923 A364350 counts combination-free strict partitions, complement A364839.
%Y A365923 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365923 A365924 counts incomplete partitions, ranks A365830, strict A365831.
%Y A365923 Cf. A000009, A006827, A122768, A364272, A365919, A365921.
%K A365923 nonn,tabl
%O A365923 0,7
%A A365923 _Gus Wiseman_, Sep 24 2023