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Regular triangle where T(n,k) is the number of antichains covering n vertices with k connected components.
8

%I #5 Aug 04 2018 06:41:36

%S 1,1,1,5,3,1,84,23,6,1,6348,470,65,10,1,7743728,39598,1575,145,15,1,

%T 2414572893530,54354104,144403,4095,280,21,1,56130437190053299918162,

%U 19316801997024,218033088,402073,9100,490,28,1

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

%H Gus Wiseman, <a href="/A048143/a048143_4.txt">Sequences enumerating clutters, antichains, hypertrees, and hyperforests, organized by labeling, spanning, and allowance of singletons</a>.

%e Triangle begins:

%e 1

%e 1 1

%e 5 3 1

%e 84 23 6 1

%e 6348 470 65 10 1

%e 7743728 39598 1575 145 15 1

%t blg={1,1,5,84,6348,7743728,2414572893530,56130437190053299918162} (*A048143*);

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Sum[Product[blg[[Length[s]]],{s,spn}],{spn,Select[sps[Range[n]],Length[#]==k&]}],{n,Length[blg]},{k,n}]

%Y First column is A048143. Row sums are A006126.

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

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

%K nonn,tabl

%O 1,4

%A _Gus Wiseman_, Aug 03 2018