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

0,5

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.

References

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

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

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

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

Crossrefs

Sequence in context: A234746 A359648 A237765 * A126009 A301282 A246063

Adjacent sequences: A249629 A249630 A249631 * A249633 A249634 A249635

Keyword

nonn,tabl

Author

Geoffrey Critzer, Nov 02 2014

Status

approved