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 A368099 #24 Jan 11 2024 20:03:57
%S A368099 1,0,1,0,2,1,0,3,4,1,0,5,12,5,1,0,7,28,22,5,1,0,11,66,83,31,5,1,0,15,
%T A368099 134,252,147,34,5,1,0,22,280,726,620,203,35,5,1,0,30,536,1946,2283,
%U A368099 1069,235,35,5,1,0,42,1043,4982,7890,5019,1469,248,35,5,1
%N A368099 Triangle read by rows where T(n,k) is the number of non-isomorphic k-element sets of finite nonempty multisets with cardinalities summing to n, or strict multiset partitions of weight n and length k.
%H A368099 Andrew Howroyd, <a href="/A368099/b368099.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e A368099 Triangle begins:
%e A368099 1
%e A368099 0 1
%e A368099 0 2 1
%e A368099 0 3 4 1
%e A368099 0 5 12 5 1
%e A368099 0 7 28 22 5 1
%e A368099 0 11 66 83 31 5 1
%e A368099 0 15 134 252 147 34 5 1
%e A368099 0 22 280 726 620 203 35 5 1
%e A368099 0 30 536 1946 2283 1069 235 35 5 1
%e A368099 0 42 1043 4982 7890 5019 1469 248 35 5 1
%e A368099 ...
%e A368099 Row n = 4 counts the following representatives:
%e A368099 . {{1,1,1,1}} {{1},{1,1,1}} {{1},{2},{1,1}} {{1},{2},{3},{4}}
%e A368099 {{1,1,1,2}} {{1},{1,1,2}} {{1},{2},{1,2}}
%e A368099 {{1,1,2,2}} {{1},{1,2,2}} {{1},{2},{1,3}}
%e A368099 {{1,1,2,3}} {{1},{1,2,3}} {{1},{2},{3,3}}
%e A368099 {{1,2,3,4}} {{1},{2,2,2}} {{1},{2},{3,4}}
%e A368099 {{1},{2,2,3}}
%e A368099 {{1},{2,3,4}}
%e A368099 {{1,1},{1,2}}
%e A368099 {{1,1},{2,2}}
%e A368099 {{1,1},{2,3}}
%e A368099 {{1,2},{1,3}}
%e A368099 {{1,2},{3,4}}
%t A368099 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A368099 mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
%t A368099 brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
%t A368099 Table[Length[Union[brute /@ Select[mpm[n],UnsameQ@@#&&Length[#]==k&]]], {n,0,5},{k,0,n}]
%o A368099 (PARI)
%o A368099 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A368099 permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o A368099 K(q, t, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
%o A368099 G(n)={my(s=0); forpart(q=n, my(p=sum(t=1, n, y^t*subst(x*Ser(K(q, t, n\t))/t, x, x^t))); s+=permcount(q)*exp(p-subst(subst(p, x, x^2), y, y^2))); s/n!}
%o A368099 T(n)={[Vecrev(p) | p <- Vec(G(n))]}
%o A368099 { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 11 2024
%Y A368099 Row sums are A316980, connected case A319557.
%Y A368099 For multiset partitions we have A317533, connected A322133.
%Y A368099 Counting connected components instead of edges gives A321194.
%Y A368099 For normal multiset partitions we have A330787, row sums A317776.
%Y A368099 For set multipartitions we have A334550.
%Y A368099 For set-systems we have A368096, row-sums A283877 (connected A300913).
%Y A368099 A000110 counts set-partitions, non-isomorphic A000041.
%Y A368099 A003465 counts covering set-systems, unlabeled A055621.
%Y A368099 A007716 counts non-isomorphic multiset partitions, connected A007718.
%Y A368099 A049311 counts non-isomorphic set multipartitions, connected A056156.
%Y A368099 A058891 counts set-systems, unlabeled A000612, connected A323818.
%Y A368099 Cf. A255903, A296122, A302545, A306005, A317532, A317775, A317794, A317795, A319560, A368094, A368095.
%K A368099 nonn,tabl
%O A368099 0,5
%A A368099 _Gus Wiseman_, Dec 31 2023