A220233 - OEIS

%I #12 Feb 13 2022 20:07:57

%S 0,0,2,6,0,3,12,24,0,4,80,60,60,0,5,390,480,180,120,0,6,2352,2730,

%T 1680,420,210,0,7,15176,18816,10920,4480,840,336,0,8,106416,136584,

%U 84672,32760,10080,1512,504,0,9,801450,1064160,682920,282240,81900,20160,2520,720,0,10

%N 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.

%C Row sums = A052512 for n>1. Column for k=0 is A220232.

%F E.g.f.: x*(exp(x*(exp(x) -1 + y)) - 1 + y)  (letting T(1,1)=1).

%e Triangle T(n,k) begins:

%e      0;

%e      0,    2;

%e      6,    0,    3;

%e     12,   24,    0,   4;

%e     80,   60,   60,   0,   5;

%e    390,  480,  180, 120,   0, 6;

%e   2352, 2730, 1680, 420, 210, 0, 7;

%e   ...

%t 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

%K nonn,tabl

%O 1,3

%A _Geoffrey Critzer_, Dec 08 2012