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A380252 Triangular array read by rows: T(n,k) is the number of labeled acyclic digraphs on n vertices with exactly k weakly connected components, n>=0, 0<=k<=n.

%I A380252 #19 Jan 18 2025 09:35:58
%S A380252 1,0,1,0,2,1,0,18,6,1,0,446,84,12,1,0,26430,2590,240,20,1,0,3596762,
%T A380252 175200,8970,540,30,1,0,1111506858,26568374,678930,24010,1050,42,1,0,
%U A380252 774460794326,9127077036,112393736,2007600,54740,1848,56,1,0,1206342801843750,7057099207134,42191272116,357391608,5013540,111636,3024,72,1
%N A380252 Triangular array read by rows: T(n,k) is the number of labeled acyclic digraphs on n vertices with exactly k weakly connected components, n>=0, 0<=k<=n.
%F A380252 E.g.f.: exp(y*log(B(x))) where B(x) = Sum_{n>=0} A003024(n)*x^n/n!.
%e A380252 Triangle T(n,k) begins:
%e A380252   1;
%e A380252   0,       1;
%e A380252   0,       2,      1;
%e A380252   0,      18,      6,    1;
%e A380252   0,     446,     84,   12,   1;
%e A380252   0,   26430,   2590,  240,  20,  1;
%e A380252   0, 3596762, 175200, 8970, 540, 30, 1;
%e A380252   ...
%t A380252 nn = 8; B[n_] := n! 2^Binomial[n, 2];e[x_] := Sum[x^n/B[n], {n, 0, nn}];egf[ggf_] := Normal[Series[ggf, {x, 0, nn}]] /.Table[x^i -> x^i*2^Binomial[i, 2], {i, 0, nn}]; Table[Drop[(Table[n!, {n, 0, nn}] CoefficientList[Series[Exp[y (Log[egf[1/e[-x]]])], {x, 0, nn}], {x,y}])[[i]], {i + 1, nn + 1}], {i, 1, nn + 1}] // Grid
%Y A380252 Columns k=0-1 give: A000007, A082402.
%Y A380252 Row sums give A003024.
%Y A380252 Cf. A082403.
%K A380252 nonn,tabl
%O A380252 0,5
%A A380252 _Geoffrey Critzer_, Jan 17 2025