A317674 Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.

1, 1, 1, 5, 3, 1, 84, 23, 6, 1, 6348, 470, 65, 10, 1, 7743728, 39598, 1575, 145, 15, 1, 2414572893530, 54354104, 144403, 4095, 280, 21, 1, 56130437190053299918162, 19316801997024, 218033088, 402073, 9100, 490, 28, 1

Offset: 1

Author

Gus Wiseman, Aug 03 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins:
        1
        1       1
        5       3       1
       84      23       6       1
     6348     470      65      10       1
  7743728   39598    1575     145      15       1
		

Links

• Gus Wiseman, Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons.

Crossrefs

First column is A048143. Row sums are A006126.

Cf. A000272, A001187, A013922, A030019, A134954, A143543, A275307, A293510.

Cf. A317634, A317635, A317671, A317672, A317677.

Programs

Mathematica

blg={1,1,5,84,6348,7743728,2414572893530,56130437190053299918162} (*A048143*);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[Product[blg[[Length[s]]],{s,spn}],{spn,Select[sps[Range[n]],Length[#]==k&]}],{n,Length[blg]},{k,n}]