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A335987 Triangular array read by rows: T(n,k) is the number of labeled quasi-orders on [n] that are composed of exactly k irreducible components n>=0, 0<=k<=n.

%I A335987 #83 Jan 06 2023 13:48:29
%S A335987 1,0,1,0,2,2,0,11,12,6,0,147,112,72,24,0,3412,1910,1020,480,120,0,
%T A335987 121553,52184,21870,9600,3600,720,0,6353629,2101540,693672,254520,
%U A335987 96600,30240,5040,0,476850636,120988214,31163496,9289728,3116400,1048320,282240,40320
%N A335987 Triangular array read by rows: T(n,k) is the number of labeled quasi-orders on [n] that are composed of exactly k irreducible components n>=0, 0<=k<=n.
%H A335987 Alois P. Heinz, <a href="/A335987/b335987.txt">Rows n = 0..18, flattened</a>
%F A335987 E.g.f.: 1/(1 - y*(1 - 1/A(x))) where A(x) is the e.g.f. for A000798.
%e A335987   1;
%e A335987   0,    1;
%e A335987   0,    2,    2;
%e A335987   0,   11,   12,    6;
%e A335987   0,  147,  112,   72,  24;
%e A335987   0, 3412, 1910, 1020, 480, 120;
%e A335987   ...
%t A335987 nn = 9; A[x_] := Total[Cases[Import["https://oeis.org/A000798/b000798.txt",
%t A335987       "Table"], {_, _}][[All, 2]]*Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];
%t A335987 Table[Take[(Range[0, nn]! CoefficientList[Series[1/(1 - y (1 - 1/A[x])), {x, 0, nn}], {x, y}])[[i]],  i], {i, 1, nn}] // Grid
%Y A335987 Cf. A000798 (row sums), A046912 (column k=1), A000142 (main diagonal), A354615.
%K A335987 nonn,tabl
%O A335987 0,5
%A A335987 _Geoffrey Critzer_, Jul 10 2022