A000405 -id:A000405 - cp's OEIS frontend

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

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

A000406 Coefficients of iterated exponentials.

Original entry on oeis.org

1, 6, 57, 741, 12244, 245755, 5809875, 158198200, 4877852505, 168055077875, 6400217406500, 267058149580823, 12118701719205803, 594291742526530761, 31323687504696772151, 1766116437541895988303, 106080070002238888908150

Offset: 1

N. J. A. Sloane

nonn

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).

T. D. Noe, Table of n, a(n) for n=1..100
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]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 302

Cf. A003713, A000268, A000310, A000359, A001765.

With[{nn=20},CoefficientList[Series[-Log[1+Log[1+Log[1+Log[1+ Log[1+ Log[1- x]]]]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 03 2015 *)

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, 6); \\ Seiichi Manyama, Feb 11 2022

my(N=20, x='x+O('x^N)); Vec(serlaplace(-log(1+log(1+log(1+log(1+log(1+log(1-x)))))))) \\ Seiichi Manyama, Feb 11 2022

E.g.f.: -log(1+log(1+log(1+log(1+log(1+log(1-x)))))).

A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

Original entry on oeis.org

1, 2, 15, 234, 6170, 245755, 13761937, 1030431500, 99399019626, 12003835242090, 1773907219147800, 314880916127332489, 66109411013740671200, 16204039283106534720952, 4585484528618722750937783, 1483746673734716952089913364, 544359300175753347889146067840

Offset: 1

Ilya Gutkovskiy, Apr 06 2018

nonn

			The initial coefficients of successive iterations of e.g.f. A(x) = -log(1 - x) are as follows:
n = 1: 0, (1), 1,   2,    6,    24,  ... e.g.f. A(x)
n = 2: 0,  1, (2),  7,   35,   228,  ... e.g.f. A(A(x))
n = 3: 0,  1,  3, (15), 105,   947,  ... e.g.f. A(A(A(x)))
n = 4: 0,  1,  4,  26, (234), 2696,  ... e.g.f. A(A(A(A(x))))
n = 5: 0,  1,  5,  40,  440, (6170), ... e.g.f. A(A(A(A(A(x)))))

Seiichi Manyama, Table of n, a(n) for n = 1..246
Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
N. J. A. Sloane, Transforms

Cf. A000268, A000310, A000359, A000406, A001765, A003713, A104150, A139383, A158832, A174482, A261280.

g:= x-> -log(1-x):
a:= n-> n! * coeff(series((g@@n)(x), x, n+1), x, n):
seq(a(n), n=1..19);  # Alois P. Heinz, Feb 11 2022

Table[n! SeriesCoefficient[Nest[Function[x, -Log[1 - x]], x, n], {x, 0, n}], {n, 17}]

T(n, k) = if(k==1, (n-1)!, sum(j=1, n, abs(stirling(n, j, 1))*T(j, k-1)));
a(n) = T(n, n); \\ Seiichi Manyama, Feb 11 2022

a(n) = T(n,n), T(n,k) = Sum_{j=1..n} |Stirling1(n,j)| * T(j,k-1), k>1, T(n,1) = (n-1)!. - Seiichi Manyama, Feb 11 2022

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

Original entry on oeis.org

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

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

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

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A000406 Coefficients of iterated exponentials.

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A302358 a(n) = coefficient of x^n in the n-th iteration (n-fold self-composition) of e.g.f. -log(1 - x).

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

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A153277 Array read by antidiagonals of higher order Bell numbers.

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