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A354615 Triangular array read by rows: T(n,k) is the number of labeled posets on [n] that are composed of exactly k irreducible posets, n >= 0, 0 <= k <= n.

%I A354615 #15 Jul 10 2022 08:26:00
%S A354615 1,0,1,0,1,2,0,7,6,6,0,97,62,36,24,0,2251,1110,510,240,120,0,80821,
%T A354615 30902,11340,4440,1800,720,0,4305127,1273566,369726,119280,42000,
%U A354615 15120,5040,0,332273257,75831422,17192196,4476024,1335600,433440,141120,40320
%N A354615 Triangular array read by rows: T(n,k) is the number of labeled posets on [n] that are composed of exactly k irreducible posets, n >= 0, 0 <= k <= n.
%F A354615 E.g.f.: 1/(1-y*(1-1/A(x))) where A(x) is the e.g.f. for A001035.
%e A354615   1;
%e A354615   0,    1;
%e A354615   0,    1,    2;
%e A354615   0,    7,    6,   6;
%e A354615   0,   97,   62,  36,  24;
%e A354615   0, 2251, 1110, 510, 240, 120;
%e A354615   ...
%t A354615 nn = 9; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",
%t A354615   "Table"], {_, _}][[All, 2]]*   Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]]; Table[Take[(Range[0, nn]!* CoefficientList[ Series[1/(1 - y (1 - 1/A[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]
%Y A354615 Cf. A046908 (column k=1), A001035 (row sums), A000142 (main diagonal).
%K A354615 nonn,tabl
%O A354615 0,6
%A A354615 _Geoffrey Critzer_, Jul 08 2022