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 A365545 #6 Sep 25 2023 12:55:53
%S A365545 1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,0,0,2,0,1,0,1,0,0,2,0,1,0,1,0,0,0,3,0,
%T A365545 1,0,0,1,1,0,0,3,0,1,0,0,0,3,0,0,0,4,0,1,0,1,0,0,2,2,0,0,4,0,1,0,1,0,
%U A365545 0,0,5,0,0,0,5,0,1,0,2,0,0,0,0,5,2,0,0,5,0,1,0
%N A365545 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.
%C A365545 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.
%C A365545 Is column k = n - 7 given by A325695?
%e A365545 Triangle begins:
%e A365545 1
%e A365545 1 0
%e A365545 0 1 0
%e A365545 1 0 1 0
%e A365545 0 1 0 1 0
%e A365545 0 0 2 0 1 0
%e A365545 1 0 0 2 0 1 0
%e A365545 1 0 0 0 3 0 1 0
%e A365545 0 1 1 0 0 3 0 1 0
%e A365545 0 0 3 0 0 0 4 0 1 0
%e A365545 1 0 0 2 2 0 0 4 0 1 0
%e A365545 1 0 0 0 5 0 0 0 5 0 1 0
%e A365545 2 0 0 0 0 5 2 0 0 5 0 1 0
%e A365545 2 0 1 0 0 0 8 0 0 0 6 0 1 0
%e A365545 1 1 3 0 0 0 0 7 3 0 0 6 0 1 0
%e A365545 2 0 4 0 1 0 0 0 12 0 0 0 7 0 1 0
%e A365545 1 1 2 2 3 1 0 0 0 11 3 0 0 7 0 1 0
%e A365545 2 0 3 0 7 0 1 0 0 0 16 0 0 0 8 0 1 0
%e A365545 3 0 0 2 6 3 3 1 0 0 0 15 4 0 0 8 0 1 0
%e A365545 Row n = 12: counts the following partitions:
%e A365545 (6,3,2,1) . . . . (9,2,1) (6,5,1) . . (11,1) . (12) .
%e A365545 (5,4,2,1) (8,3,1) (6,4,2) (10,2)
%e A365545 (7,4,1) (9,3)
%e A365545 (7,3,2) (8,4)
%e A365545 (5,4,3) (7,5)
%t A365545 Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Complement[Range[n], Total/@Subsets[#]]]==k&]],{n,0,10},{k,0,n}]
%Y A365545 Row sums are A000009, non-strict A000041.
%Y A365545 The complement (positive subset-sums) is also A365545 with rows reversed.
%Y A365545 Weighted row sums are A365922, non-strict A365918.
%Y A365545 The non-strict version is A365923, complement A365658, rank stat A325799.
%Y A365545 A046663 counts partitions without a subset summing to k, strict A365663.
%Y A365545 A126796 counts complete partitions, ranks A325781, strict A188431.
%Y A365545 A364350 counts combination-free strict partitions, complement A364839.
%Y A365545 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365545 A365924 counts incomplete partitions, ranks A365830, strict A365831.
%Y A365545 Cf. A006827, A284640, A304792, A364272, A365921.
%K A365545 nonn,tabl
%O A365545 0,18
%A A365545 _Gus Wiseman_, Sep 24 2023