A144150 -id:A144150 - cp's OEIS frontend

A111672 Array T(n,k) = A153277(n-1,k) = A144150(n,k-1) read by downwards antidiagonals.

1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 12, 15, 1, 1, 5, 22, 60, 52, 1, 1, 6, 35, 154, 358, 203, 1, 1, 7, 51, 315, 1304, 2471, 877, 1, 1, 8, 70, 561, 3455, 12915, 19302, 4140, 1

Offset: 1

Gary W. Adamson, Aug 14 2005

Column k is obtained by taking the k-th matrix power of the triangle A008277 and multiplying from the right with the column vector [1,0,0,0,....].

			The array starts
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,  ...

Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 4.

Cf. A008277, A144150, A153277.

Cf. A000326 (row 3), A005945 (row 4), A000110 (column 2), A000258 (column 3), A000307 (column 4), A000357 (column 5), A000405 (column 6), A111669 (column 7), A081624.

a(44) and definition corrected by Georg Fischer, May 18 2022

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

Original entry on oeis.org

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

A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 6, 5, 1, 0, 1, 1, 5, 10, 14, 7, 1, 0, 1, 1, 6, 15, 30, 27, 11, 1, 0, 1, 1, 7, 21, 55, 75, 58, 15, 1, 0, 1, 1, 8, 28, 91, 170, 206, 111, 22, 1, 0, 1, 1, 9, 36, 140, 336, 571, 518, 223, 30, 1, 0

Offset: 0

Alois P. Heinz, Jul 28 2017

A(n,k) is the number of unlabeled rooted trees with exactly n leaves, all in level k.  A(3,3) = 6:
:    o        o       o        o        o       o
:    |        |       |       / \      / \     /|\
:    o        o       o      o   o    o   o   o o o
:    |       / \     /|\     |   |   ( )  |   | | |
:    o      o   o   o o o    o   o   o o  o   o o o
:   /|\    ( )  |   | | |   ( )  |   | |  |   | | |
:  o o o   o o  o   o o o   o o  o   o o  o   o o o

			Square array A(n,k) begins:
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  1, 1,  1,   1,    1,    1,     1,     1,      1, ...
  0, 1,  2,   3,    4,    5,     6,     7,      8, ...
  0, 1,  3,   6,   10,   15,    21,    28,     36, ...
  0, 1,  5,  14,   30,   55,    91,   140,    204, ...
  0, 1,  7,  27,   75,  170,   336,   602,   1002, ...
  0, 1, 11,  58,  206,  571,  1337,  2772,   5244, ...
  0, 1, 15, 111,  518, 1789,  5026, 12166,  26328, ...
  0, 1, 22, 223, 1344, 5727, 19193, 54046, 133476, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
B. A. Huberman and T. Hogg, Complexity and adaptation, Evolution, games and learning (Los Alamos, N.M., 1985). Phys. D 22 (1986), no. 1-3, 376-384.
Index entries for sequences related to rooted trees

Columns k=1-10 give: A000012, A000041, A001970, A007713, A007714, A290355, A290356, A290357, A290358, A290359.
Rows 0+1,2-10 give: A000012, A001477, A000217, A000330, A007715, A290360, A290361, A290362, A290363, A290364.
Main diagonal gives A290354.
Cf. A144150.

with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, 1, `if`(k=0, 0, add(
      add(A(d, k-1)*d, d=divisors(j))*A(n-j, k), j=1..n)/n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_]:=b[n, k]=If[n<2, 1, If[k==0, 0, Sum[Sum[A[d, k - 1]*d, {d, Divisors[j]}] A[n - j, k], {j, n}]/n]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}]//Flatten (* Indranil Ghosh, Jul 30 2017, after Maple code *)

G.f. of column k=0: 1+x, of column k>0: Product_{j>0} 1/(1-x^j)^A(j,k-1).

A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 33, 21, 8, 1, 11, 91, 104, 36, 10, 1, 15, 298, 452, 238, 55, 12, 1, 22, 910, 2335, 1430, 455, 78, 14, 1, 30, 3017, 11992, 10179, 3505, 775, 105, 16, 1, 42, 9945, 66810, 74299, 31881, 7297, 1218, 136, 18, 1, 56

Offset: 1

Gus Wiseman, Jan 27 2019

			Array begins:
      k=1:  k=2:  k=3:  k=4:  k=5:  k=6:
  n=1:  1     1     1     1     1     1
  n=2:  2     4     6     8    10    12
  n=3:  3    10    21    36    55    78
  n=4:  5    33   104   238   455   775
  n=5:  7    91   452  1430  3505  7297
  n=6: 11   298  2335 10179 31881 80897
Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:
  {{111}}          {{112}}          {{123}}
  {{1}{11}}        {{1}{12}}        {{1}{23}}
  {{1}}{{11}}      {{2}{11}}        {{1}}{{23}}
  {{1}{1}{1}}      {{1}}{{12}}      {{1}{2}{3}}
  {{1}}{{1}{1}}    {{1}{1}{2}}      {{1}}{{2}{3}}
  {{1}}{{1}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}
                   {{1}}{{1}{2}}
                   {{2}}{{1}{1}}
                   {{1}}{{1}}{{2}}

Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).
Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).
Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Apply[Rule,Table[{undats[m][[i]],i},{i,Length[undats[m]]}],{1}]],First[Sort[expnorm[m,1]]]]];
expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#1>=aft&]}]},Union@@(expnorm[#1,aft+1]&)/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]]]];
strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];
kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];
Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]

a(46)-a(56) from Robert Price, May 11 2021

A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 6, 5, 1, 1, 1, 5, 10, 13, 7, 1, 1, 1, 6, 15, 26, 24, 11, 1, 1, 1, 7, 21, 45, 59, 48, 15, 1, 1, 1, 8, 28, 71, 120, 140, 86, 22, 1, 1, 1, 9, 36, 105, 216, 326, 307, 160, 30, 1, 1, 1, 10, 45, 148, 357, 657, 835, 684, 282, 42, 1

Offset: 0

Paul D. Hanna, Jul 07 2004

Main diagonal forms A096752. Antidiagonal sums form A096753. Row with index n lists the row sums of the n-th matrix power of triangle A096651, for n>=0.

			n-th row lists n-dimensional partitions; table begins with n=0:
  [1,1,1,1,1,1,1,1,1,1,1,1,...],
  [1,1,2,3,5,7,11,15,22,30,42,56,...],
  [1,1,3,6,13,24,48,86,160,282,500,859,...],
  [1,1,4,10,26,59,140,307,684,1464,3122,...],
  [1,1,5,15,45,120,326,835,2145,5345,...],
  [1,1,6,21,71,216,657,1907,5507,15522,...],
  [1,1,7,28,105,357,1197,3857,12300,38430,...],
  [1,1,8,36,148,554,2024,7134,24796,84625,...],
  [1,1,9,45,201,819,3231,12321,46209,170370,...],
  [1,1,10,55,265,1165,4927,20155,80920,...],...
Array begins:
      k=0:  k=1:  k=2:  k=3:  k=4:  k=5:  k=6:  k=7:  k=8:
  n=0:  1     1     1     1     1     1     1     1     1
  n=1:  1     1     2     3     5     7    11    15    22
  n=2:  1     1     3     6    13    24    48    86   160
  n=3:  1     1     4    10    26    59   140   307   684
  n=4:  1     1     5    15    45   120   326   835  2145
  n=5:  1     1     6    21    71   216   657  1907  5507
  n=6:  1     1     7    28   105   357  1197  3857 12300
  n=7:  1     1     8    36   148   554  2024  7134 24796
  n=8:  1     1     9    45   201   819  3231 12321 46209
  n=9:  1     1    10    55   265  1165  4927 20155 80920

G. E. Andrews, The Theory of Partitions, Add.-Wes. 1976, pp. 189-197.

Pontus von Brömssen, Table of n, a(n) for n = 0..275 (first 23 antidiagonals)
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]

Rows: A000012 (n=0), A000041 (n=1), A000219 (n=2), A000293 (n=3), A000334 (n=4), A000390 (n=5), A000416 (n=6), A000427 (n=7), A179855 (n=8).
Columns: A008778 (k=4), A008779 (k=5), A042984 (k=6).
Cf. A096651, A096752, A096753.
Cf. A096806.
Cf. A042984.
Cf. A144150, A213427, A290353, A323718, A323719.

trans[x_]:=If[x=={},{},Transpose[x]];
levptns[n_,k_]:=If[k==1,IntegerPartitions[n],Join@@Table[Select[Tuples[levptns[#,k-1]&/@y],And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#,ConstantArray[n,k-1]]&/@#)])&],{y,IntegerPartitions[n]}]];
Table[If[sum==k,1,Length[levptns[k,sum-k]]],{sum,0,10},{k,0,sum}] (* Gus Wiseman, Jan 27 2019 *)

T(0, n)=T(n, 0)=T(n, 1)=1 for n>=0.

Inverse binomial transforms of the columns is given by triangle A096806.

A000307 Number of 4-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 4, 22, 154, 1304, 12915, 146115, 1855570, 26097835, 402215465, 6734414075, 121629173423, 2355470737637, 48664218965021, 1067895971109199, 24795678053493443, 607144847919796830, 15630954703539323090, 421990078975569031642, 11918095123121138408128

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.
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..440
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 293
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.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, J. Phys. A: Math.Gen 37 (2004) 3475-3487.
John Riordan, Letter, Apr 28 1976.
Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Index entries for sequences related to rooted trees

a(n)=|A039812(n,1)| (first column of triangle).
Cf. A000110, A000258, A000357, A000405, A001669.
Column k=3 of A144150.

g:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, (n-1)! *add(p(k)*b(n-k)/ (k-1)!/ (n-k)!, k=1..n)) end end: a:= g(g(g(1))): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 11 2008

nn = 18; a = Exp[Exp[x] - 1]; b = Exp[a - 1];
Range[0, nn]! CoefficientList[Series[Exp[b - 1], {x, 0, nn}], x]  (*Geoffrey Critzer, Dec 28 2011*)

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

a(n) = sum(sum(sum(stirling2(n,k) *stirling2(k,m) *stirling2(m,r), k=m..n), m=r..n), r=1..n), n>0. - Vladimir Kruchinin, Sep 08 2010

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

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

A189233 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals upwards, where the e.g.f. of column k is exp(k*(e^x-1)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 22, 12, 4, 1, 0, 52, 94, 57, 20, 5, 1, 0, 203, 454, 309, 116, 30, 6, 1, 0, 877, 2430, 1866, 756, 205, 42, 7, 1, 0, 4140, 14214, 12351, 5428, 1555, 330, 56, 8, 1, 0, 21147, 89918, 88563, 42356, 12880, 2850, 497, 72, 9, 1

Offset: 0

Peter Luschny, Apr 18 2011

A(n,k) is the n-th moment of a Poisson distribution with mean = k. - Geoffrey Critzer, Dec 23 2018

			Square array begins:
       A000007 A000110 A001861 A027710 A078944 A144180 A144223 A144263
A000012   1,    1,    1,    1,    1,     1,     1,     1, ...
A001477   0,    1,    2,    3,    4,     5,     6,     7, ...
A002378   0,    2,    6,   12,   20,    30,    42,    56, ...
A033445   0,    5,   22,   57,  116,   205,   330,   497, ...
          0,   15,   94,  309,  756,  1555,  2850,  4809, ...
          0,   52,  454, 1866, 5428, 12880, 26682, 50134, ...

Vincenzo Librandi, Table of n, a(n) for n = 0..5150
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Peter Luschny, Set partitions and Bell numbers

Columns: A000007, A000110, A001861, A027710, A078944, A144180, A144223, A144263.
Rows: A000012, A001477, A002378, A033445.
Main diagonal gives A242817.
Cf. A000587, A144150.

# Cf. also the Maple prog. of Alois P. Heinz in A144223 and A144180.
expnums := proc(k,n) option remember; local j;
`if`(n = 0, 1, (1+add(binomial(n-1,j-1)*expnums(k,n-j), j = 1..n-1))*k) end:
A189233_array := (k, n) -> expnums(k,n):
seq(print(seq(A189233_array(k,n), k = 0..7)), n = 0..5);
A189233_egf := k -> exp(k*(exp(x)-1));
T := (n,k) -> n!*coeff(series(A189233_egf(k), x, n+1), x, n):
seq(lprint(seq(T(n,k), k = 0..7)), n = 0..5):
# alternative Maple program:
A:= proc(n, k) option remember; `if`(n=0, 1,
      (1+add(binomial(n-1, j-1)*A(n-j, k), j=1..n-1))*k)
    end:
seq(seq(A(d-k, k), k=0..d), d=0..12);  # Alois P. Heinz, Sep 25 2017

max = 9; Clear[col]; col[k_] := col[k] = CoefficientList[ Series[ Exp[k*(Exp[x]-1)], {x, 0, max}], x]*Range[0, max]!; a[0, ] = 1; a[n?Positive, 0] = 0; a[n_, k_] := col[k][[n+1]]; Table[ a[n-k, k], {n, 0, max}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)
Table[Table[BellB[n, k], {k, 0, 5}], {n, 0, 5}] // Grid  (* Geoffrey Critzer, Dec 23 2018 *)

A(n,k):=if k=0 and n=0 then 1 else if k=0 then 0 else  sum(stirling2(n,i)*k^i,i,0,n); /* Vladimir Kruchinin, Apr 12 2019 */

E.g.f. of column k: exp(k*(e^x-1)).
A(n,1) = A000110(n), A(n, -1) = A000587(n).
A(n,k) = BellPolynomial(n, k). - Geoffrey Critzer, Dec 23 2018
A(n,k) = Sum_{i=0..n} Stirling2(n,i)*k^i. - Vladimir Kruchinin, Apr 12 2019

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

cp's OEIS Frontend

A111672 Array T(n,k) = A153277(n-1,k) = A144150(n,k-1) read by downwards antidiagonals.

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

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A290353 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the k-th Euler transform of the sequence with g.f. 1+x.

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A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

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A096751 Square table, read by antidiagonals, where T(n,k) equals the number of n-dimensional partitions of k.

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A000307 Number of 4-level labeled rooted trees with n leaves.

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

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