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.

2, 6, 3, 36, 24, 4, 320, 240, 60, 5, 3750, 3000, 900, 120, 6, 54432, 45360, 15120, 2520, 210, 7, 941192, 806736, 288120, 54880, 5880, 336, 8, 18874368, 16515072, 6193152, 1290240, 161280, 12096, 504, 9, 430467210, 382637520, 148803480, 33067440, 4592700, 408240, 22680, 720

Offset: 2

Author

Geoffrey Critzer, Feb 07 2012

Keywords

• nonn
• tabl

Examples

			Triangle begins:
      2;
      6     3;
     36    24     4;
    320   240    60    5;
   3750  3000   900  120    6;
  54432 45360 15120 2520  210  7;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
• Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 179.

Crossrefs

Column 1 is A055541.

Row sums are A000169.

Programs

Mathematica

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

PARI

T(n)={my(f=serreverse(x*exp(-x + O(x^n)))); [Vecrev(p/y) | p<-Vec(serlaplace(x*exp(y*f) - x))]}
{ my(A=T(7)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 22 2020

Formula

E.g.f.: x*exp(y * T(x)) where T(x) is the e.g.f. for A000169.