A039812 -id:A039812 - cp's OEIS frontend

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

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

A039813 Matrix 5th power of Stirling2 triangle A008277.

Original entry on oeis.org

1, 5, 1, 35, 15, 1, 315, 215, 30, 1, 3455, 3325, 725, 50, 1, 44590, 56605, 17100, 1825, 75, 1, 660665, 1060780, 415555, 60900, 3850, 105, 1, 11035095, 21772595, 10606470, 1998605, 172550, 7210, 140, 1, 204904830, 486459105, 286281665, 66528210, 7346955, 417690, 12390, 180, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins:
      1;
      5,     1;
     35,    15,     1;
    315,   215,    30,    1;
   3455,  3325,   725,   50,  1;
  44590, 56605, 17100, 1825, 75, 1;
  ...

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

Cf. A008277, A000357 (first column).

Cf. A039810, A039811, A039812.

max = 9; m = MatrixPower[Array[StirlingS2, {max, max}], 5]; Table[Take[m[[n]], n], {n, 1, max}] // Flatten (* Jean-François Alcover, Mar 03 2014 *)

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

A039811 Triangle read by rows: matrix cube of the Stirling2 triangle A008277.

Original entry on oeis.org

1, 3, 1, 12, 9, 1, 60, 75, 18, 1, 358, 660, 255, 30, 1, 2471, 6288, 3465, 645, 45, 1, 19302, 65051, 47838, 12495, 1365, 63, 1, 167894, 728556, 685580, 235193, 35700, 2562, 84, 1, 1606137, 8792910, 10285488, 4444188, 877653, 86940, 4410, 108, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins
     1;
     3,    1;
    12,    9,    1;
    60,   75,   18,   1;
   358,  660,  255,  30,  1;
  2471, 6288, 3465, 645, 45, 1;
  ...

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

Cf. A008277, A000258 (first column).

Cf. also A039810, A039812, A039813.

Flatten[Table[SeriesCoefficient[(Exp[Exp[Exp[x]-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(x)-1)-1)-1 )^k)/k!. [corrected by Seiichi Manyama, Feb 12 2022]

A039816 Triangle read by rows: matrix 4th power of the Stirling-1 triangle A008275.

Original entry on oeis.org

1, -4, 1, 26, -12, 1, -234, 152, -24, 1, 2696, -2210, 500, -40, 1, -37919, 36976, -10710, 1240, -60, 1, 630521, -704837, 245896, -36750, 2590, -84, 1, -12111114, 15132932, -6120324, 1109696, -101500, 4816, -112, 1, 264051201, -362099010, 165387680, -34990620, 3901296, -241164, 8232, -144, 1

Offset: 1

Christian G. Bower, Feb 15 1999

			Triangle begins:
       1;
      -4,     1;
      26,   -12,      1;
    -234,   152,    -24,    1;
    2696, -2210,    500,  -40,   1;
  -37919, 36976, -10710, 1240, -60, 1;
  ...

Seiichi Manyama, Rows n = 1..140, flattened
Gabriella Bretti, Pierpaolo Natalini and Paolo E. Ricci, A new set of Sheffer-Bell polynomials and logarithmic numbers, Georgian Mathematical Journal, Feb. 2019, page 9.

Cf. A000310 (first column), A008275.

Cf. A039812, A039814, A039815, A039817.

T:= Matrix(10,10,(i,j) -> `if`(i>= j, combinat:-stirling1(i,j),0)):
M:= T^4:
seq(seq(M[i,j],j=1..i),i=1..10); # Robert Israel, Sep 12 2022

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

E.g.f. of k-th column: ((log(1+log(1+log(1+log(1+x)))))^k)/k!.

A351514 Expansion of e.g.f. (exp(exp(exp(exp(x)-1)-1)-1)-1)^2 / 2.

Original entry on oeis.org

1, 12, 136, 1650, 21904, 318521, 5051988, 86910426, 1612648066, 32107793135, 682724688430, 15439016490989, 369914992674530, 9359103270641290, 249292192469843244, 6971850327184526783, 204215496402215939638, 6251233458455082035922

Offset: 2

Seiichi Manyama, Feb 12 2022

nonn

Column 2 of A039812.

Cf. A000307, A000558, A351513, A351515.

my(N=20, x='x+O('x^N)); Vec(serlaplace((exp(exp(exp(exp(x)-1)-1)-1)-1)^2/2))

T(n, k) = if(k==0, n<=1, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = sum(k=1, n-1, binomial(n-1, k)*T(k, 4)*T(n-k, 4));

a(n) = Sum_{k=1..n-1} binomial(n-1,k) * A000307(k) * A000307(n-k).

cp's OEIS Frontend

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

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A039813 Matrix 5th power of Stirling2 triangle A008277.

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A039811 Triangle read by rows: matrix cube of the Stirling2 triangle A008277.

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A039816 Triangle read by rows: matrix 4th power of the Stirling-1 triangle A008275.

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A351514 Expansion of e.g.f. (exp(exp(exp(exp(x)-1)-1)-1)-1)^2 / 2.

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