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 A365381 #6 Sep 08 2023 22:53:05
%S A365381 1,2,1,4,2,2,1,8,4,4,5,2,2,1,16,8,8,10,10,7,5,5,2,2,1,32,16,16,20,20,
%T A365381 23,15,15,12,12,8,5,5,2,2,1,64,32,32,40,40,46,47,38,33,35,29,28,21,17,
%U A365381 14,13,8,5,5,2,2,1,128,64,64,80,80,92,94,102,79,82,76,75,68,64,53,48,43,34,33,23,19,15,13,8,5,5,2,2,1
%N A365381 Irregular triangle read by rows where T(n,k) is the number of subsets of {1..n} with a subset summing to k.
%C A365381 Row lengths are A000124(n) = 1 + n*(n+1)/2.
%e A365381 Triangle begins:
%e A365381 1
%e A365381 2 1
%e A365381 4 2 2 1
%e A365381 8 4 4 5 2 2 1
%e A365381 16 8 8 10 10 7 5 5 2 2 1
%e A365381 32 16 16 20 20 23 15 15 12 12 8 5 5 2 2 1
%e A365381 64 32 32 40 40 46 47 38 33 35 29 28 21 17 14 13 8 5 5 2 2 1
%e A365381 Array begins:
%e A365381 k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9
%e A365381 -------------------------------------------------------
%e A365381 n=0: 1
%e A365381 n=1: 2 1
%e A365381 n=2: 4 2 2 1
%e A365381 n=3: 8 4 4 5 2 2 1
%e A365381 n=4: 16 8 8 10 10 7 5 5 2 2
%e A365381 n=5: 32 16 16 20 20 23 15 15 12 12
%e A365381 n=6: 64 32 32 40 40 46 47 38 33 35
%e A365381 n=7: 128 64 64 80 80 92 94 102 79 82
%e A365381 n=8: 256 128 128 160 160 184 188 204 207 184
%e A365381 n=9: 512 256 256 320 320 368 376 408 414 440
%e A365381 The T(5,8) = 12 subsets are:
%e A365381 {3,5} {1,2,5} {1,2,3,4} {1,2,3,4,5}
%e A365381 {1,3,4} {1,2,3,5}
%e A365381 {1,3,5} {1,2,4,5}
%e A365381 {2,3,5} {1,3,4,5}
%e A365381 {3,4,5} {2,3,4,5}
%t A365381 Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#],k]&]],{n,0,8},{k,0,n*(n+1)/2}]
%Y A365381 Row lengths are A000124 = number of distinct sums of subsets of {1..n}.
%Y A365381 Central column/main diagonal is A365376.
%Y A365381 A000009 counts sets summing to n.
%Y A365381 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365381 A365046 counts combination-full subsets, differences of A364914.
%Y A365381 Cf. A007865, A085489, A093971, A103580, A131577, A151897, A326080, A364272, A364534, A365073, A365377, A365380.
%K A365381 nonn,tabf
%O A365381 0,2
%A A365381 _Gus Wiseman_, Sep 08 2023