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A206429 Triangular array read by rows. T(n,k) is the number of rooted labeled trees on n nodes such that the root node has degree k. n>=2, 1<=k<=n-1.

%I A206429 #23 Jan 22 2020 21:01:32
%S A206429 2,6,3,36,24,4,320,240,60,5,3750,3000,900,120,6,54432,45360,15120,
%T A206429 2520,210,7,941192,806736,288120,54880,5880,336,8,18874368,16515072,
%U A206429 6193152,1290240,161280,12096,504,9,430467210,382637520,148803480,33067440,4592700,408240,22680,720
%N A206429 Triangular array read by rows. T(n,k) is the number of rooted labeled trees on n nodes such that the root node has degree k. n>=2, 1<=k<=n-1.
%H A206429 Andrew Howroyd, <a href="/A206429/b206429.txt">Table of n, a(n) for n = 2..1276</a> (first 50 rows)
%H A206429 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 179.
%F A206429 E.g.f.: x*exp(y * T(x)) where T(x) is the e.g.f. for A000169.
%e A206429 Triangle begins:
%e A206429       2;
%e A206429       6     3;
%e A206429      36    24     4;
%e A206429     320   240    60    5;
%e A206429    3750  3000   900  120    6;
%e A206429   54432 45360 15120 2520  210  7;
%t A206429 nn=10;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Map[f,Drop[Transpose[Table[Range[0,nn]!CoefficientList[Series[x t^k/k!,{x,0,nn}],x],{k,1,8}]],2]]//Flatten
%o A206429 (PARI) T(n)={my(f=serreverse(x*exp(-x + O(x^n)))); [Vecrev(p/y) | p<-Vec(serlaplace(x*exp(y*f) - x))]}
%o A206429 { my(A=T(7)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 22 2020
%Y A206429 Column 1 is A055541.
%Y A206429 Row sums are A000169.
%K A206429 nonn,tabl
%O A206429 2,1
%A A206429 _Geoffrey Critzer_, Feb 07 2012