A249632 Triangular array read by rows. T(n,k) is the number of labeled trees with black and white nodes having exactly k black nodes, n>=0, 0<=k<=n.

1, 1, 1, 1, 2, 1, 3, 9, 9, 3, 16, 64, 96, 64, 16, 125, 625, 1250, 1250, 625, 125, 1296, 7776, 19440, 25920, 19440, 7776, 1296, 16807, 117649, 352947, 588245, 588245, 352947, 117649, 16807, 262144, 2097152, 7340032, 14680064, 18350080, 14680064, 7340032, 2097152, 262144

Offset: 0

Author

Geoffrey Critzer, Nov 02 2014

Keywords

• nonn
• tabl

Comments

Row sums = A038058.

T(n,n) = T(n,0) = n^(n-2) free trees A000272.

T(n,n-1) = T(n,1) = n^(n-1) rooted trees A000169.

T(n,2) = A081131.

Examples

			1,
1,    1,
1,    2,    1,
3,    9,    9,     3,
16,   64,   96,    64,    16,
125,  625,  1250,  1250,  625,   125,
1296, 7776, 19440, 25920, 19440, 7776, 1296
		

References

• F. Harary and E. Palmer, Graphical Enumeration, Academic Press,1973, page 30, exercise 1.10.

Programs

Mathematica

nn = 6; f[x_] := Sum[n^(n - 2) x^n/n!, {n, 1, nn}];
Map[Select[#, # > 0 &] &,
  Range[0, nn]! CoefficientList[
    Series[f[x + y x] + 1, {x, 0, nn}], {x, y}]] // Grid

Formula

E.g.f.: A(x + y*x) where A(x) is the e.g.f. for A000272.