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A343882 Triangular array read by rows: T(n,k) is the number of transitive relations on n labeled nodes with exactly k connected components.

%I A343882 #15 May 05 2021 20:29:59
%S A343882 1,0,2,0,9,4,0,109,54,8,0,2647,1115,216,16,0,110481,36280,6790,720,32,
%T A343882 0,7291543,1801927,287475,32020,2160,64,0,726434549,133060816,
%U A343882 16873619,1718290,129080,6048,128,0,106312974249,14380028959,1387285830,118346473,8584240,467488,16128,256
%N A343882 Triangular array read by rows: T(n,k) is the number of transitive relations on n labeled nodes with exactly k connected components.
%C A343882 T(n,n) = 2^n since each node is reflexive or not.
%H A343882 Geoffrey Critzer, <a href="/A343882/a343882.pdf">Finite Topologies</a>
%F A343882 E.g.f.: A(x + exp(x) -1)^y where A(x) is the e.g.f. for A001035.
%e A343882 Triangular array T(n,k) begins:
%e A343882   1;
%e A343882   0,      2;
%e A343882   0,      9,     4;
%e A343882   0,    109,    54,    8;
%e A343882   0,   2647,  1115,  216,  16;
%e A343882   0, 110481, 36280, 6790, 720, 32;
%e A343882   ...
%t A343882 A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt", "Table"], {_, _}][[All, 2]]* Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]]; nn = 10;
%t A343882 Range[0, nn]! CoefficientList[ Series[Exp[y Log[A[ x + Exp[ x] - 1]]], {x, 0, nn}], {x,y}] // Grid;Table[Take[(Range[0, nn]! CoefficientList[Series[Exp[y Log[A[ x + Exp[ x] - 1]]], {x, 0, nn}], {x, y}])[[i, All]], i], {i, 1, nn}] // Grid
%t A343882   (* Import function in code after _Jean-François Alcover_ *)
%Y A343882 Cf. A245731 (column k=1), A006905 (row sums), A001035.
%K A343882 nonn,tabl
%O A343882 0,3
%A A343882 _Geoffrey Critzer_, May 02 2021