A000307 -id:A000307 - cp's OEIS frontend

A000258 Expansion of e.g.f. exp(exp(exp(x)-1)-1).

1, 1, 3, 12, 60, 358, 2471, 19302, 167894, 1606137, 16733779, 188378402, 2276423485, 29367807524, 402577243425, 5840190914957, 89345001017415, 1436904211547895, 24227076487779802, 427187837301557598, 7859930038606521508, 150601795280158255827

Offset: 0

N. J. A. Sloane

Number of 3-level labeled rooted trees with n leaves. - Christian G. Bower, Aug 15 1998
Number of pairs of set partitions (d,d') of [n] such that d is finer than d'. - A. Joseph Kennedy (kennedy_2001in(AT)yahoo.co.in), Feb 05 2006
In the Comm. Algebra paper cited, I introduce a sequence of algebras called 'class partition algebras' with this sequence as dimensions. The algebras are the centralizers of wreath product in combinatorial representation theory. - A. Joseph Kennedy (kennedy_2001in(AT)yahoo.co.in), Feb 17 2008
a(n) is the number of ways to partition {1,2,...,n} and then partition each cell (block) into subcells.

			G.f. = 1 + x + 3*x^2 + 12*x^3 + 60*x^4 + 358*x^5 + 2471*x^6 + 19302*x^7 + ...

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.4.

Alois P. Heinz, Table of n, a(n) for n = 0..475 (first 101 terms from T. D. Noe)
A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Brauer and Jones tied monoids, arXiv:2107.04170 [math.RT], 2021.
P. Blasiak, A. Horzela, K. A. Penson, G. H. E. Duchamp and A. I. Solomon, Boson normal ordering via substitutions and Sheffer-type polynomials, arXiv:quant-ph/0501155, 2005.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
P. J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
Gottfried Helms, Bell Numbers, 2008.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 70
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 292
A. Joseph Kennedy, Class partition algebras as centralizer algebras, Communications in Algebra, 35 (2007), 145-170, see page 153.
A. Joseph Kennedy, P. Jaish, and P. Sundaresan, Note on generating function of higher dimensional bell numbers (sic), Malaya Journal of Matematik (2020) Vol.8, No. 2, 369-372.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
T. Mansour, A. Munagi, and M. Shattuck, Recurrence Relations and Two-Dimensional Set Partitions , J. Int. Seq. 14 (2011) #11.4.1.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003, [J. Phys. A 37 (2004), 3475-3487].
John Riordan, Letter, Apr 28 1976.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003, [Order 21 (2004), 83-89].
Index entries for sequences related to rooted trees

Cf. A000110, A000307, A000357, A000405, A001669, A039811.
Row sums of (Stirling2)^2 triangle A130191.
Column k=2 of A144150.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(Exp(x)-1)-1))); [Factorial(n-1)*b[n]: n in [1..m]]; // Vincenzo Librandi, Feb 01 2020

with(combinat, bell, stirling2): seq(add(stirling2(n,k)*(bell(k)), k=0..n),n=0..30);
with(combstruct); SetSetSetL := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},labeled];
# alternative Maple program:
b:= proc(n, t) option remember; `if`(n=0 or t=0, 1, add(
       b(n-j, t)*b(j, t-1)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 02 2021

nn = 20; Range[0, nn]! CoefficientList[Series[Exp[Exp[Exp[x] - 1] - 1], {x, 0, nn}], x]
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp[ Exp[x] - 1] - 1] , {x, 0, n}]]; (* Michael Somos, Aug 15 2015 *)
a[n_] := Sum[StirlingS2[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 06 2016 *)
Table[Sum[BellY[n, k, BellB[Range[n]]], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

makelist(sum(stirling2(n,k)*belln(k),k,0,n),n,0,24); /* Emanuele Munarini, Jul 04 2011 */

a(n) = |A039811(n, 1)| (first column of triangle).
a(n) = Sum_{k=0..n} Stirling2(n, k)*Bell(k). - Detlef Pauly (dettodet(AT)yahoo.de), Jun 06 2002
Representation as an infinite series (Dobinski-type formula), in Maple notation: a(n)=exp(exp(-1)-1)*sum(evalf(sum(p!*stirling2(k, p)*exp(-p), p=1..k))*k^n/k!, k=0..infinity), n=1, 2, ... . - Karol A. Penson, Nov 28 2003
a(n) = Sum_{k=0..n} A055896(n,k). - R. J. Mathar, Apr 15 2008
G.f.: Sum_{j>=0} Bell(j)*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019

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.

Original entry on oeis.org

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.

A000357 Number of 5-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 5, 35, 315, 3455, 44590, 660665, 11035095, 204904830, 4183174520, 93055783320, 2238954627848, 57903797748386, 1601122732128779, 47120734323344439, 1470076408565099152, 48449426629560437576, 1681560512531504058350, 61293054886119796799892

Offset: 0

N. J. A. Sloane

J. de la Cal, J. Carcamo, Set partitions and moments of random variables, J. Math. Anal. Applic. 378 (2011) 16 doi:10.1016/j.jmaa.2011.01.002 Remark 5
J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..400
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
Gottfried Helms, Bell Numbers, 2008.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 294
Index entries for sequences related to rooted trees

a(n)=|A039813(n,1)| (first column of triangle).
Cf. A000110, A000258, A000307, A000405, A001669.
Column k=4 of A144150.

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:= g(g(g(g(1)))): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 11 2008

max = 17; Join[{1}, MatrixPower[Array[StirlingS2, {max, max}], 5][[All, 1]]] (* Jean-François Alcover, Mar 03 2014 *)

E.g.f.: exp(exp(exp(exp(exp(x)-1)-1)-1)-1).

Extended with new description by Christian G. Bower, Aug 15 1998

A000405 Number of 6-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 6, 51, 561, 7556, 120196, 2201856, 45592666, 1051951026, 26740775306, 742069051906, 22310563733864, 722108667742546, 25024187820786357, 924161461265888370, 36223781285638309482, 1501552062016443881514

Offset: 0

N. J. A. Sloane

J. de la Cal, J. Carcamo, Set partitions and moments of random variables, J. Math. Anal. Applic. 378 (2011) 16 doi:10.1016/j.jmaa.2011.01.002 Remark 5
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..150
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
Gottfried Helms, Bell Numbers, 2008.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 295
Index entries for sequences related to rooted trees

Cf. A000110, A000258, A000307, A000357, A001669.

Column k=5 of A144150.

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:= g(g(g(g(g(1))))): seq(a(n), n=0..30); # Alois P. Heinz, Sep 11 2008

g[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (n-1)!*Sum[p[k]*b[n-k]/(k-1)!/(n-k)!, {k, 1, n}]]; b]; a = Nest[g, 1&, 5]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 12 2014, after Alois P. Heinz *)

E.g.f.: exp(exp(exp(exp(exp(exp(x)-1)-1)-1)-1)-1).

Extended with new definition by Christian G. Bower, Aug 15 1998

A001669 Number of 7-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 7, 70, 910, 14532, 274778, 5995892, 148154860, 4085619622, 124304629050, 4133867297490, 149114120602860, 5796433459664946, 241482353893283349, 10730629952953517859, 506500241174366575122, 25302666611855946733140

Offset: 0

N. J. A. Sloane

J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..150
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 296
Index entries for sequences related to rooted trees

Cf. A000110, A000258, A000307, A000357, A000405.

Column k=6 of A144150.

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:= g(g(g(g(g(g(1)))))): seq(a(n), n=0..30); # Alois P. Heinz, Sep 11 2008

g[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (n-1)!*Sum[p[k]*b[n-k]/(k-1)!/(n-k)!, {k, 1, n}]]; b]; a = Nest[g, 1&, 6]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
With[{nn=20},Join[{1},Rest[CoefficientList[Series[Nest[Exp[#]-1&,Exp[x]-1,6],{x,0,nn}],x] Range[0,nn]!]]] (* Harvey P. Dale, Mar 02 2015 *)

E.g.f.: exp(exp(exp(exp(exp(exp(exp(x)-1)-1)-1)-1)-1)-1).

Extended with new definition by Christian G. Bower, Aug 15 1998

A050338 Number of ways of factoring n with 2 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 4, 1, 4, 1, 10, 4, 4, 1, 16, 1, 4, 4, 30, 1, 16, 1, 16, 4, 4, 1, 54, 4, 4, 10, 16, 1, 22, 1, 75, 4, 4, 4, 74, 1, 4, 4, 54, 1, 22, 1, 16, 16, 4, 1, 176, 4, 16, 4, 16, 1, 54, 4, 54, 4, 4, 1, 102, 1, 4, 16, 206, 4, 22, 1, 16, 4, 22, 1, 267, 1, 4, 16, 16, 4, 22, 1, 176, 30, 4, 1, 102

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

			4 = ((4)) = ((2*2)) = ((2)*(2)) = ((2))*((2)).

R. J. Mathar, Table of n, a(n) for n = 1..1679

Cf. A001055, A050336-A050341. a(p^k)=A007713. a(A002110)=A000307.

Dirichlet g.f.: Product_{n>=2}(1/(1-1/n^s)^A050336(n)).

a(n) = A050339(A101296(n)). - R. J. Mathar, May 26 2017

A139383 Number of n-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 2, 12, 154, 3455, 120196, 5995892, 406005804, 35839643175, 3998289746065, 550054365477936, 91478394767427823, 18091315306315315610, 4196205472500769304318, 1128136777063831105273242, 347994813261017613045578964, 122080313159891715442898099217

Offset: 0

Paul D. Hanna, Apr 16 2008

nonn

Define the matrix function matexps(M) to be exp(M)/exp(1). Then the number of k-level labeled rooted trees with n leaves is also column 0 of the triangle resulting from the n-th iteration of matexps on the Pascal matrix P, A007318. The resulting triangle is also S^n*P*S^-n, where S is the Stirling2 matrix A048993. This function can be coded in PARI as sum(k=0,200,1./k!*M^k)/exp(1), using exp(M) does not work. See A056857, which equals (1/e)*exp(P) or S*P*S^-1. - Gerald McGarvey, Aug 19 2009

			If we form a table from the family of sequences defined by:
number of k-level labeled rooted trees with n leaves,
then this sequence equals the diagonal in that table:
n=1:A000012=[1,1,1,1,1,1,1,1,1,1,...];
n=2:A000110=[1,2,5,15,52,203,877,4140,21147,115975,...];
n=3:A000258=[1,3,12,60,358,2471,19302,167894,1606137,...];
n=4:A000307=[1,4,22,154,1304,12915,146115,1855570,26097835,...];
n=5:A000357=[1,5,35,315,3455,44590,660665,11035095,204904830,...];
n=6:A000405=[1,6,51,561,7556,120196,2201856,45592666,1051951026,...];
n=7:A001669=[1,7,70,910,14532,274778,5995892,148154860,4085619622,...];
n=8:A081624=[1,8,92,1380,25488,558426,14140722,406005804,13024655442,...];
n=9:A081629=[1,9,117,1989,41709,1038975,29947185,979687005,35839643175,..].
Row n in the above table equals column 0 of matrix power A008277^n where A008277 = triangle of Stirling numbers of 2nd kind:
1;
1,1;
1,3,1;
1,7,6,1;
1,15,25,10,1;
1,31,90,65,15,1; ...
The name of this sequence is a generalization of the definition given in the above sequences by _Christian G. Bower_.

Alois P. Heinz, Table of n, a(n) for n = 0..247

Cf. A008277; A000110, A000258, A000307, A000357, A000405, A001669, A081624, A081629, A261280, A290354.

A diagonal of A144150.

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:
a:= n-> A(n, n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 14 2015
# second Maple program:
g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1+(g@@n)(x), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 31 2017
# third Maple program:
b:= proc(n, t, m) option remember; `if`(t=0, `if`(n<2, 1, 0),
     `if`(n=0, b(m, t-1, 0), m*b(n-1, t, m)+b(n-1, t, m+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021

t[n_,m_]:=t[n,m] = If[m==1,1,Sum[StirlingS2[n,k]*t[k,m-1],{k,1,n}]]; Table[t[n,n],{n,1,20}] (* Vaclav Kotesovec, Aug 14 2015 after Vladimir Kruchinin *)

T(n,m):=if m=1 then 1 else sum(stirling2(n,i)*T(i,m-1),i,1,n);
makelist(T(n,n),n,1,7); /* Vladimir Kruchinin, May 19 2012 */

{a(n)=local(E=exp(x+x*O(x^n))-1,F=x); for(i=1,n,F=subst(F,x,E));n!*polcoeff(F,n)}

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))
def a(n): return A(n, n - 1)
print([a(n) for n in range(21)]) # Indranil Ghosh, Aug 07 2017, after Maple code

a(n) = T(n,n), T(n,m) = Sum_{i=1..n} Stirling2(n,i)*T(i,m-1), m>1, T(n,1)=1. - Vladimir Kruchinin, May 19 2012
a(n) = n! * [x^n] 1 + g^n(x), where g(x) = exp(x)-1. - Alois P. Heinz, Aug 14 2015
From Vaclav Kotesovec, Aug 14 2015: (Start)
Conjecture: a(n) ~ c * n^(2*n-5/6) / (2^(n-1) * exp(n)), where c = 2.86539...
a(n) ~ exp(-1) * A261280(n).
(End)

a(0)=1 prepended by Alois P. Heinz, Jul 31 2017

A039812 Triangle read by rows: matrix 4th power of the Stirling2 triangle A008277.

Original entry on oeis.org

1, 4, 1, 22, 12, 1, 154, 136, 24, 1, 1304, 1650, 460, 40, 1, 12915, 21904, 8550, 1160, 60, 1, 146115, 318521, 162904, 30590, 2450, 84, 1, 1855570, 5051988, 3246068, 789824, 86940, 4592, 112, 1, 26097835, 86910426, 68151304, 20606796, 2919504, 210924, 7896, 144, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins
      1;
      4,     1;
     22,    12,    1;
    154,   136,   24,    1;
   1304,  1650,  460,   40,  1;
  12915, 21904, 8550, 1160, 60, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened

Cf. A008277, A000307 (first column).

Cf. A039810, A039811, A039813.

Flatten[Table[SeriesCoefficient[(Exp[Exp[Exp[Exp[x]-1]-1]-1]-1)^k, {x,0,n}]  n!/k!, {n,9}, {k,n}]] (* Stefano Spezia, Sep 12 2022 *)

E.g.f. k-th column: (( exp(exp(exp(exp(x)-1)-1)-1)-1 )^k)/k!. [corrected by Seiichi Manyama, Feb 12 2022]

cp's OEIS Frontend

A000258 Expansion of e.g.f. exp(exp(exp(x)-1)-1).

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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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A000357 Number of 5-level labeled rooted trees with n leaves.

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A000405 Number of 6-level labeled rooted trees with n leaves.

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A001669 Number of 7-level labeled rooted trees with n leaves.

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A050338 Number of ways of factoring n with 2 levels of parentheses.

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A139383 Number of n-level labeled rooted trees with n leaves.

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