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A228890 Triangular array read by rows. T(n,k) is the number of 2-colored labeled graphs on n nodes with exactly k edges; n >= 0, 0 <= k <= A002620(n).

%I A228890 #27 Mar 28 2023 08:01:09
%S A228890 1,2,4,2,8,12,6,16,48,60,32,6,32,160,360,440,310,120,20,64,480,1680,
%T A228890 3480,4680,4212,2520,960,210,20,128,1344,6720,20720,43680,66108,73514,
%U A228890 60480,36540,15820,4662,840,70,256,3584,24192,103040,308560,686784,1172976,1565888,1649340,1373680,900592,459312,178416,50960,10080,1232,70
%N A228890 Triangular array read by rows.  T(n,k) is the number of 2-colored labeled graphs on n nodes with exactly k edges; n >= 0, 0 <= k <= A002620(n).
%C A228890 A 2-colored labeled graph is a simple labeled graph in which each vertex is painted black or white and black vertices are only connected to white vertices and vice versa. [corrected by _Geoffrey Critzer_, Mar 27 2023]
%F A228890 E.g.f.: Sum_{n>=0} exp(1 + y)^n*x^n/n!
%e A228890 Triangle begins:
%e A228890    1;
%e A228890    2;
%e A228890    4,   2;
%e A228890    8,  12,    6;
%e A228890   16,  48,   60,   32,    6;
%e A228890   32, 160,  360,  440,  310,  120,   20;
%e A228890   64, 480, 1680, 3480, 4680, 4212, 2520, 960, 210, 20;
%e A228890   ...
%t A228890 nn=6;f[x_,y_]:=Sum[Exp[x (1+y)^n]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[f[x,y],{x,0,nn}],{x,y}]]//Grid
%Y A228890 Row sums are A047863.
%Y A228890 Column k=0 gives A000079.
%Y A228890 Cf. A002620.
%K A228890 nonn,tabf
%O A228890 0,2
%A A228890 _Geoffrey Critzer_, Sep 07 2013