A235595 Triangle read by rows: the triangle in A034855, with the n-th row normalized by dividing it by n.

1, 1, 2, 1, 9, 6, 1, 40, 60, 24, 1, 195, 560, 420, 120, 1, 1056, 5550, 6240, 3240, 720, 1, 6321, 59472, 94710, 68880, 27720, 5040, 1, 41392, 692440, 1527456, 1426320, 792960, 262080, 40320, 1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880, 1, 2237920, 119723130, 490458240, 691331760, 570810240, 323114400, 125798400, 30844800, 3628800

Offset: 2

Author

N. J. A. Sloane, Jan 14 2014

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of forests of labeled rooted trees with n nodes and height k Cf. A210725. Equivalently, T(n,k) is the number of nilpotent partial functions on [n] with index k+1. - Geoffrey Critzer, Nov 26 2021

Examples

			Triangle begins:
1.
1, 2,
1, 9, 6,
1, 40, 60, 24,
1, 195, 560, 420, 120,
1, 1056, 5550, 6240, 3240, 720,
1, 6321, 59472, 94710, 68880, 27720, 5040,
1, 41392, 692440, 1527456,1426320, 792960, 262080, 40320,
1, 293607, 8753040, 26418168, 30560544, 21213360, 9676800, 2721600, 362880,
...
		

Links

• Alois P. Heinz, Rows n = 2..142, flattened

Crossrefs

Cf. A000435, A034855, A001854, A210725, A234953, A235596, A236396.

Row sums give A000272.

Programs

Maple

b:= proc(n, h) option remember; `if`(min(n, h)=0, 1, add(
      binomial(n-1, j-1)*j*b(j-1, h-1)*b(n-j, h), j=1..n))
    end:
T:= (n,k)-> b(n-1, k-1)-b(n-1, k-2):
seq(seq(T(n, d), d=1..n-1), n=2..12);  # Alois P. Heinz, Aug 21 2017

Mathematica

gf[k_] := gf[k] = If[k == 0, x, x*E^gf[k-1]]; a[n_, k_] := n!*Coefficient[Series[gf[k], {x, 0, n+1}], x, n]; t[n_, k_] := (a[n, k] - a[n, k-1])/n; Table[t[n, k], {n, 2, 11}, {k, 1, n-1}] // Flatten (* Jean-François Alcover, Mar 18 2014, after Alois P. Heinz *)

Python

from sympy import binomial
from sympy.core.cache import cacheit
@cacheit
def b(n, h): return 1 if min(n, h)==0 else sum([binomial(n - 1, j - 1)*j*b(j - 1, h - 1)*b(n - j, h) for j in range(1, n + 1)])
def T(n, k): return b(n - 1, k - 1) - b(n - 1, k - 2)
for n in range(2, 13): print([T(n, d) for d in  range(1, n)]) # Indranil Ghosh, Aug 26 2017, after Maple code

Formula

A234953(n) = Sum_{k=1..n} k*T(n,k).