A210725 Triangle read by rows: T(n,k) = number of forests of labeled rooted trees with n nodes and height at most k (n>=1, 0<=k<=n-1).

1, 1, 3, 1, 10, 16, 1, 41, 101, 125, 1, 196, 756, 1176, 1296, 1, 1057, 6607, 12847, 16087, 16807, 1, 6322, 65794, 160504, 229384, 257104, 262144, 1, 41393, 733833, 2261289, 3687609, 4480569, 4742649, 4782969, 1, 293608, 9046648, 35464816, 66025360, 87238720, 96915520, 99637120, 100000000

Offset: 1

Author

N. J. A. Sloane, May 09 2012

Keywords

• nonn
• tabl

Examples

			Triangle begins:
  1;
  1,    3;
  1,   10,   16;
  1,   41,  101,   125;
  1,  196,  756,  1176,  1296;
  1, 1057, 6607, 12847, 16087, 16807;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• J. Riordan, Forests of labeled trees, J. Combin. Theory, 5 (1968), 90-103.

Crossrefs

Diagonals include A000248, A000949, A000950, A000951, A000272.

Programs

Maple

f:= proc(k) f(k):= `if`(k<0, 1, exp(x*f(k-1))) end:
T:= (n, k)-> coeff(series(f(k), x, n+1), x, n) *n!:
seq(seq(T(n, k), k=0..n-1), n=1..9); # Alois P. Heinz, May 30 2012
# second Maple program:
T:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*j*T(j-1, h-1)*T(n-j, h), j=1..n))
    end:
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Aug 21 2017

Mathematica

f[?Negative] = 1; f[k] := Exp[x*f[k-1]]; t[n_, k_] := Coefficient[Series[f[k], {x, 0, n+1}], x, n]*n!; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 9}] // Flatten (* Jean-François Alcover, Oct 30 2013, after Maple *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def T(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*T(j - 1, h - 1)*T(n - j, h) for j in range(1, n + 1)])
for n in range(1, 11): print([T(n, k) for k in range(n)]) # Indranil Ghosh, Aug 21 2017, after second Maple code