A111672 -id:A111672 - cp's OEIS frontend

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

Alois P. Heinz, Sep 11 2008

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

			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,  ...

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

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.

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

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 *)

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

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.

A153277 Array read by antidiagonals of higher order Bell numbers.

Original entry on oeis.org

1, 1, 2, 1, 3, 5, 1, 4, 12, 15, 1, 5, 22, 60, 52, 1, 6, 35, 154, 358, 203, 1, 7, 51, 315, 1304, 2471, 877, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1, 9, 92, 910, 7556, 44590, 146115, 167894, 21147, 1, 10, 117, 1380, 14532, 120196, 660665, 1855570, 1606137, 115975

Offset: 1

Jonathan Vos Post, Dec 22 2008

Mezo's abstract: The powers of matrices with Stirling number-coefficients are investigated. It is revealed that the elements of these matrices have a number of properties of the ordinary Stirling numbers. Moreover, "higher order" Bell, Fubini and Eulerian numbers can be defined. Hence we give a new interpretation for E. T. Bell's iterated exponential integers. In addition, it is worth to note that these numbers appear in combinatorial physics, in the problem of the normal ordering of quantum field theoretical operators.

			The table on p.4 of Mezo begins:
===========================================================
B_p,n|n=1|n=2|n=3.|.n=4.|..n=5.|....n=6.|.....n=7.|comment
===========================================================
p=1..|.1.|.2.|..5.|..15.|...52.|....203.|.....877.|.A000110
p=2..|.1.|.3.|.12.|..60.|..358.|...2471.|...19302.|.A000258
p=3..|.1.|.4.|.22.|.154.|.1304.|..12915.|..146115.|.A000307
p=4..|.1.|.5.|.35.|.315.|.3455.|..44590.|..660665.|.A000357
p=5..|.1.|.6.|.51.|.561.|.7556.|.120196.|.2201856.|.A000405
===========================================================

E. T. Bell, The iterated exponential integers, Ann. Math. 39(3) (1938), 539-557.
J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
Istvan Mezo, On powers of Stirling matrices, arXiv:0812.4047.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela, A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, J.Phys. A: Math.Gen. 37 3475-3487 (2004).

Cf. A000110, A000258, A000307, A000357, A000405, A111672.
From Alois P. Heinz, Feb 02 2009: (Start)
Truncated and reflected version of A144150.
Cf. A001669, A081624, A081629, A081697, A081740, A000326, A005945. (End)

g:= proc(a) local b; b:=proc(n) option remember; if n=0 then 1 else (n-1)! *add (a(k)* b(n-k)/ (k-1)!/ (n-k)!, k=1..n) fi end end: B:= (p,n)-> (g@@p)(1)(n):
seq(seq(B(d-n, n), n=1..d-1), d=1..12); # Alois P. Heinz, Feb 02 2009

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-n+1], {k, 1, 12}, {n, 1, k}] // Flatten (* Jean-François Alcover, Jan 28 2015 *)

More terms from Alois P. Heinz, Feb 02 2009

A209631 Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to identity function, evaluated at k.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 10, 4, 1, 1, 5, 20, 41, 5, 1, 1, 6, 33, 127, 196, 6, 1, 1, 7, 49, 280, 967, 1057, 7, 1, 1, 8, 68, 518, 2883, 8549, 6322, 8, 1, 1, 9, 90, 859, 6689, 34817, 85829, 41393, 9, 1, 1, 10, 115, 1321, 13310, 101841, 481477

Offset: 0

Peter Luschny, Mar 11 2012

Motivation: The exponential transform applied n times to the constant function 1 evaluated at k was studied by E. T. Bell (see A144150).

			n\k [0][1][2] [3]   [4]    [5]     [6]
[0]  0, 1, 2,  3,    4,     5,      6
[1]  1, 1, 3, 10,   41,   196,   1057   [A000248]
[2]  1, 1, 4, 20,  127,   967,   8549   [A007550]
[3]  1, 1, 5, 33,  280,  2883,  34817
[4]  1, 1, 6, 49,  518,  6689, 101841
[5]  1, 1, 7, 68,  859, 13310, 243946
[6]  1, 1, 8, 90, 1321, 23851, 510502
column3(n) = (3*n^2 + 11*n + 6)/2!
column4(n) = (18*n^3 + 93*n^2 + 111*n + 24)/3!
column5(n) = (180*n^4 + 1180*n^3 + 2160*n^2 + 1064*n + 120)/4!
column6(n) = (2700*n^5+21225*n^4+51850*n^3+41835*n^2+8510*n+720)/5!

Discussion on seqcomp: A little challenge

Cf. A000248, A007550, A111672, A144150.

# Implementation after Alois P. Heinz.
exptr := proc(p) local g; g := proc(n) option remember; local k;
`if`(n=0, 1, add(binomial(n-1, k-1)*p(k)*g(n-k), k=1..n)) end end:
A209631 := (n,k) -> (exptr@@n)(m->m)(k):
seq(lprint(seq(A209631(n,k), k=0..6)), n=0..6);

exptr[p_] := Module[{g}, g[n_] := g[n] = Module[{k}, If[n == 0, 1, Sum[Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n}]]]; g]; A209631[n_, k_] := Nest[exptr, Identity, n][k]; Table[A209631[n-k , k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 27 2014, after Alois P. Heinz *)

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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.

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A153277 Array read by antidiagonals of higher order Bell numbers.

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A209631 Square array A(n,k), n>=0, k>=0, read by antidiagonals, A(n,k) = exponential transform applied n times to identity function, evaluated at k.

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Maple

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