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 A365661 #11 Apr 05 2025 23:17:49
%S A365661 1,1,1,1,0,1,2,1,1,2,2,1,0,1,2,3,1,1,1,1,3,4,2,2,1,2,2,4,5,2,2,2,2,2,
%T A365661 2,5,6,3,2,3,1,3,2,3,6,8,3,3,4,3,3,4,3,3,8,10,5,4,5,4,3,4,5,4,5,10,12,
%U A365661 5,5,5,5,5,5,5,5,5,5,12
%N A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.
%C A365661 First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.
%C A365661 Rows are palindromic.
%C A365661 Are there only two zeros in the whole triangle?
%H A365661 Robert Price, <a href="/A365661/b365661.txt">Table of n, a(n) for n = 0..1325</a>
%e A365661 Triangle begins:
%e A365661 1
%e A365661 1 1
%e A365661 1 0 1
%e A365661 2 1 1 2
%e A365661 2 1 0 1 2
%e A365661 3 1 1 1 1 3
%e A365661 4 2 2 1 2 2 4
%e A365661 5 2 2 2 2 2 2 5
%e A365661 6 3 2 3 1 3 2 3 6
%e A365661 8 3 3 4 3 3 4 3 3 8
%e A365661 Row n = 6 counts the following strict partitions:
%e A365661 (6) (5,1) (4,2) (3,2,1) (4,2) (5,1) (6)
%e A365661 (5,1) (3,2,1) (3,2,1) (3,2,1) (3,2,1) (5,1)
%e A365661 (4,2) (4,2)
%e A365661 (3,2,1) (3,2,1)
%e A365661 Row n = 10 counts the following strict partitions:
%e A365661 A 91 82 73 64 532 64 73 82 91 A
%e A365661 64 541 532 532 541 541 541 532 532 541 64
%e A365661 73 631 721 631 631 4321 631 631 721 631 73
%e A365661 82 721 4321 721 4321 4321 721 4321 721 82
%e A365661 91 4321 4321 4321 4321 91
%e A365661 532 532
%e A365661 541 541
%e A365661 631 631
%e A365661 721 721
%e A365661 4321 4321
%t A365661 Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#],k]&]], {n,0,10},{k,0,n}]
%Y A365661 Columns k = 0 and k = n are A000009.
%Y A365661 The non-strict complement is A046663, central column A006827.
%Y A365661 Central column n = 2k is A237258.
%Y A365661 For subsets instead of partitions we have A365381.
%Y A365661 The non-strict case is A365543.
%Y A365661 The complement is A365663.
%Y A365661 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365661 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A365661 Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.
%K A365661 nonn,tabl
%O A365661 0,7
%A A365661 _Gus Wiseman_, Sep 16 2023