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 A367404 #6 Nov 18 2023 18:18:34
%S A367404 1,1,1,2,1,2,3,2,2,2,5,3,4,2,3,7,5,6,4,3,3,11,7,9,6,6,3,4,15,11,13,10,
%T A367404 9,6,4,4,22,15,20,13,15,9,8,4,5,30,22,27,21,21,15,12,8,5,5,42,30,39,
%U A367404 28,30,21,20,12,10,5,6,56,42,53,41,42,33,28,20,15,10,6,6
%N A367404 Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.
%C A367404 We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.
%e A367404 The partition y = (3,2,1,1) has semi-sum 3 = 2+1, but no semi-sum 6, so y is counted under T(7,3) but not under T(7,6).
%e A367404 Triangle begins:
%e A367404 1
%e A367404 1 1
%e A367404 2 1 2
%e A367404 3 2 2 2
%e A367404 5 3 4 2 3
%e A367404 7 5 6 4 3 3
%e A367404 11 7 9 6 6 3 4
%e A367404 15 11 13 10 9 6 4 4
%e A367404 22 15 20 13 15 9 8 4 5
%e A367404 30 22 27 21 21 15 12 8 5 5
%e A367404 42 30 39 28 30 21 20 12 10 5 6
%e A367404 56 42 53 41 42 33 28 20 15 10 6 6
%e A367404 77 56 73 55 60 42 44 28 25 15 12 6 7
%e A367404 Row n = 7 counts the following partitions:
%e A367404 (511) (421) (331) (421) (511) (61)
%e A367404 (4111) (3211) (322) (4111) (421) (52)
%e A367404 (3211) (2221) (3211) (322) (331) (43)
%e A367404 (31111) (22111) (31111) (3211)
%e A367404 (22111) (211111) (2221)
%e A367404 (211111) (22111)
%e A367404 (1111111)
%t A367404 Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#, {2}],k]&]], {n,2,10}, {k,2,n}]
%Y A367404 Column k = 0 is A000041.
%Y A367404 Column n = k is A004526.
%Y A367404 The complement for all submultisets is A046663, strict A365663.
%Y A367404 For subsets instead of partitions we have A365541, non-binary A365381.
%Y A367404 The non-binary version is A365543, strict A365661.
%Y A367404 Row sums are A366738.
%Y A367404 The strict case is A367405.
%Y A367404 Cf. A122768, A108917, A299701, A304792, A364272, A364911, A365658.
%K A367404 nonn,tabl
%O A367404 2,4
%A A367404 _Gus Wiseman_, Nov 17 2023