A000357 -id:A000357 - cp's OEIS frontend

A351428 Expansion of e.g.f. 1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1).

1, -1, -3, -11, -41, -75, 1540, 37725, 657715, 10551750, 163089430, 2407275470, 31865298262, 290682880132, -2479867505029, -267542605513289, -11438897571729494, -404343336811199242, -13192591498632627584, -410340915410006575406, -12233989907129223814578

Offset: 0

Seiichi Manyama, Feb 11 2022

sign

Column k=5 of A351429.
Cf. A000587, A130410, A351427.
Cf. A000357, A351423.

g:= x-> exp(x)-1:
a:= n-> n! * coeff(series(1/((g@@5)(x)+1), x, n+1), x, n):
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 11 2022

T[n_, 0] := (-1)^n * n!; T[n_, k_] := T[n, k] = Sum[StirlingS2[n, j]*T[j, k - 1], {j, 0, n}]; a[n_] := T[n, 5]; Array[a, 20, 0] (* Amiram Eldar, Feb 11 2022 *)
With[{nn=20},CoefficientList[Series[1/Exp[Exp[Exp[Exp[Exp[x]-1]-1]-1]-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 09 2025 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1)))

T(n, k) = if(k==0, (-1)^n*n!, sum(j=0, n, stirling(n, j, 2)*T(j, k-1)));
a(n) = T(n, 5);

a(n) = T(n,5), T(n,k) = Sum_{j=0..n} Stirling2(n,j) * T(j,k-1), k>1, T(n,0) = (-1)^n * n!.

A081697 10-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 10, 145, 2755, 64660, 1804705, 58336855, 2141867440, 87998832685, 3998289746065, 198991311832840, 10762795518750121, 628439018694857887, 39390402253060922833, 2637469071097179922603, 187848412983167698626469, 14178423030415044515701642

Offset: 0

Benoit Cloitre, Apr 23 2003

nonn

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.

Alois P. Heinz, Table of n, a(n) for n = 0..360

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

Column k=9 of A144150.

a(n)=local(X); if(n<0,0,X=x+x*O(x^n);  n!*
polcoeff(exp(exp(exp(exp(exp(exp(exp(exp(exp(exp(X)-1)-1)-1)-1)-1)-1)-1)-1)-1),n)).

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

A081740 11-level labeled rooted trees with n leaves.

Original entry on oeis.org

1, 1, 11, 176, 3696, 95986, 2967041, 106296586, 4328071506, 197304236151, 9951699489061, 550054365477936, 33053174868315877, 2144972900520659506, 149472637758381213628, 11130201727845695463914, 881841184375010602801553, 74061565980075915066583527

Offset: 0

Benoit Cloitre, Apr 23 2003

nonn

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.

Alois P. Heinz, Table of n, a(n) for n = 0..360

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

Column k=10 of A144150.

a(n)=local(X); if(n<0,0,X=x+x*O(x^n); n!*polcoeff (exp( exp( exp( exp( exp( exp(exp(exp(exp(exp(exp(X)-1)-1)-1)-1)-1)-1)-1)-1)-1)-1), n))

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

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

A351515 Expansion of e.g.f. (exp(exp(exp(exp(exp(x)-1)-1)-1)-1)-1)^2 / 2.

Original entry on oeis.org

1, 15, 215, 3325, 56605, 1060780, 21772595, 486459105, 11760431325, 305942552245, 8521928511915, 253041654671949, 7977871631560394, 266128899746035160, 9363456107172891499, 346487270686107589124, 13450341325170239245308, 546470289216642540029570

Offset: 2

Seiichi Manyama, Feb 12 2022

nonn

Column 2 of A039813.

Cf. A000357, A000558, A351513, A351514.

my(N=20, x='x+O('x^N)); Vec(serlaplace((exp(exp(exp(exp(exp(x)-1)-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, 5)*T(n-k, 5));

a(n) = Sum_{k=1..n-1} binomial(n-1,k) * A000357(k) * A000357(n-k).

A050349 Number of ways to factor n into distinct factors with 3 levels of parentheses.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 1, 5, 1, 5, 1, 15, 1, 5, 5, 11, 1, 15, 1, 15, 5, 5, 1, 45, 1, 5, 5, 15, 1, 35, 1, 25, 5, 5, 5, 65, 1, 5, 5, 45, 1, 35, 1, 15, 15, 5, 1, 130, 1, 15, 5, 15, 1, 45, 5, 45, 5, 5, 1, 145, 1, 5, 15, 60, 5, 35, 1, 15, 5, 35, 1, 240, 1, 5, 15, 15, 5, 35, 1, 130, 11, 5, 1, 145, 5

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).

			6 = (((6))) = (((3*2))) = (((3)*(2))) = (((3))*((2))) = (((3)))*(((2))).

R. J. Mathar, Table of n, a(n) for n = 1..1727

Cf. A045778, A050345-A050350. a(p^k)=A050344. a(A002110)=A000357.

Dirichlet g.f.: Product_{n>=2}(1+1/n^s)^A050347(n).

a(n) = A050350(A101296(n)). - R. J. Mathar, May 26 2017

cp's OEIS Frontend

A351428 Expansion of e.g.f. 1/exp(exp(exp(exp(exp(x)-1)-1)-1)-1).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A081697 10-level labeled rooted trees with n leaves.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Formula

A081740 11-level labeled rooted trees with n leaves.

Views

Author

Keywords

References

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A153277 Array read by antidiagonals of higher order Bell numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A351515 Expansion of e.g.f. (exp(exp(exp(exp(exp(x)-1)-1)-1)-1)-1)^2 / 2.

Views

Author

Keywords

Crossrefs

Programs

PARI

PARI

Formula

A050349 Number of ways to factor n into distinct factors with 3 levels of parentheses.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula