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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, 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
1,3
COMMENTS
Row sums = A052512 for n>1. Column for k=0 is A220232.
FORMULA
E.g.f.: x*(exp(x*(exp(x) -1 + y)) - 1 + y) (letting T(1,1)=1).
EXAMPLE
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;
...
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
CROSSREFS
Sequence in context: A232178 A016590 A079461 * A084897 A021388 A011040
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 08 2012
STATUS
approved