A144150 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where the e.g.f. of column k is 1+g^(k+1)(x) with g = x-> exp(x)-1.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 12, 15, 1, 1, 1, 5, 22, 60, 52, 1, 1, 1, 6, 35, 154, 358, 203, 1, 1, 1, 7, 51, 315, 1304, 2471, 877, 1, 1, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1, 1, 1, 9, 92, 910, 7556, 44590, 146115, 167894, 21147, 1, 1, 1, 10, 117

Offset: 0

Author

Alois P. Heinz, Sep 11 2008

Keywords

• nonn
• tabl

Comments

A(n,k) is also the number of (k+1)-level labeled rooted trees with n leaves.

Number of ways to start with set {1,2,...,n} and then repeat k times: partition each set into subsets. - Alois P. Heinz, Aug 14 2015

Equivalently, A(n,k) is the number of length k+1 multichains from bottom to top in the set partition lattice of an n-set. - Geoffrey Critzer, Dec 05 2020

Examples

			Square array begins:
  1,  1,   1,    1,    1,    1,  ...
  1,  1,   1,    1,    1,    1,  ...
  1,  2,   3,    4,    5,    6,  ...
  1,  5,  12,   22,   35,   51,  ...
  1, 15,  60,  154,  315,  561,  ...
  1, 52, 358, 1304, 3455, 7556,  ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• E. T. Bell, The Iterated Exponential Integers, Annals of Mathematics, 39(3) (1938), 539-557.
• Pierpaolo Natalini and Paolo Emilio Ricci, Higher order Bell polynomials and the relevant integer sequences, in Appl. Anal. Discrete Math. 11 (2017), 327-339.
• Pierpaolo Natalini and Paolo E. Ricci, Integer Sequences Connected with Extensions of the Bell Polynomials, Journal of Integer Sequences, 2017, Vol. 20, #17.10.2.
• Ivar Henning Skau and Kai Forsberg Kristensen, An asymptotic Formula for the iterated exponential Bell Numbers, arXiv:1903.07979 [math.CO], 2019.
• Ivar Henning Skau and Kai Forsberg Kristensen, Sets of iterated Partitions and the Bell iterated Exponential Integers, arXiv:1903.08379 [math.CO], 2019.
• Index entries for sequences related to rooted trees

Crossrefs

Columns k=0-10 give: A000012, A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A081697, A081740.

Rows n=0+1, 2-5 give: A000012, A000027, A000326, A005945, A005946.

First lower diagonal gives A139383.

First upper diagonal gives A346802.

Main diagonal gives A261280.

Cf. A000142, A111672, A290353.

Programs

Maple

g:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1
      else (n-1)! *add(p(k)*b(n-k)/(k-1)!/(n-k)!, k=1..n) fi
    end end:
A:= (n,k)-> (g@@k)(1)(n):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
      add(binomial(n-1, j-1)*A(j, k-1)*A(n-j, k), j=1..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 14 2015
# third Maple program:
b:= proc(n, t, m) option remember; `if`(t=0, 1, `if`(n=0,
      b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
    end:
A:= (n, k)-> b(n, k, 0):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 04 2021

Mathematica

g[k_] := g[k] = Nest[Function[x, E^x - 1], x, k]; a[n_, k_] := SeriesCoefficient[1 + g[k + 1], {x, 0, n}]*n!; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)

Python

from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def A(n, k): return 1 if n==0 or k==0 else sum([binomial(n - 1, j - 1)*A(j, k - 1)*A(n - j, k) for j in range(1, n + 1)])
for n in range(51): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 07 2017

Formula

E.g.f. of column k: 1 + g^(k+1)(x) with g = x-> exp(x)-1.

Column k+1 is Stirling transform of column k.