A220233 Triangular array read by rows. T(n,k) is the number of labeled rooted trees of height at most 2 with exactly k leaves at a distance 1 from the root, n>=1, 0<=k<=n-1.
0, 0, 2, 6, 0, 3, 12, 24, 0, 4, 80, 60, 60, 0, 5, 390, 480, 180, 120, 0, 6, 2352, 2730, 1680, 420, 210, 0, 7, 15176, 18816, 10920, 4480, 840, 336, 0, 8, 106416, 136584, 84672, 32760, 10080, 1512, 504, 0, 9, 801450, 1064160, 682920, 282240, 81900, 20160, 2520, 720, 0, 10
Offset: 1
Examples
Triangle T(n,k) begins:
0;
0, 2;
6, 0, 3;
12, 24, 0, 4;
80, 60, 60, 0, 5;
390, 480, 180, 120, 0, 6;
2352, 2730, 1680, 420, 210, 0, 7;
...
Programs
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Mathematica
nn=7;f[list_]:=Select[list,#>0&];a=x (Exp[x]-1+y);Prepend[Drop[Map[Insert[#,0,-2]&,Map[f,Range[0,nn]!CoefficientList[Series[x (Exp[a]-1+y),{x,0,nn}],{x,y}]]],2],{0}]//Grid
Formula
E.g.f.: x*(exp(x*(exp(x) -1 + y)) - 1 + y) (letting T(1,1)=1).
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