A033917 -id:A033917 - cp's OEIS frontend

A215703 A(n,k) is the n-th derivative of f_k at x=1, and f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 4, 3, 0, 1, 1, 2, 12, 8, 0, 1, 1, 6, 9, 52, 10, 0, 1, 1, 4, 27, 32, 240, 54, 0, 1, 1, 2, 18, 156, 180, 1188, -42, 0, 1, 1, 2, 15, 100, 1110, 954, 6804, 944, 0, 1, 1, 8, 9, 80, 650, 8322, 6524, 38960, -5112, 0, 1, 1, 6, 48, 56, 590, 4908, 70098, 45016, 253296, 47160, 0

Offset: 0

Alois P. Heinz, Aug 21 2012

A000081(m) distinct functions are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways.  Some functions are representable in more than one way, the number of valid parenthesizations is A000108(m-1).  The f_k are ordered, such that the number m of x's in f_k is a nondecreasing function of k.  The exact ordering is defined by the algorithm below.
The list of functions f_1, f_2, ... begins:
  | f_k : m : function (tree)  : representation(s)        : sequence |
  +-----+---+------------------+--------------------------+----------+
  | f_1 | 1 | x -> x           | x                        | A019590  |
  | f_2 | 2 | x -> x^x         | x^x                      | A005727  |
  | f_3 | 3 | x -> x^(x*x)     | (x^x)^x                  | A215524  |
  | f_4 | 3 | x -> x^(x^x)     | x^(x^x)                  | A179230  |
  | f_5 | 4 | x -> x^(x*x*x)   | ((x^x)^x)^x              | A215704  |
  | f_6 | 4 | x -> x^(x^x*x)   | (x^x)^(x^x), (x^(x^x))^x | A215522  |
  | f_7 | 4 | x -> x^(x^(x*x)) | x^((x^x)^x)              | A215705  |
  | f_8 | 4 | x -> x^(x^(x^x)) | x^(x^(x^x))              | A179405  |

			Square array A(n,k) begins:
  1,   1,    1,    1,     1,     1,     1,     1, ...
  1,   1,    1,    1,     1,     1,     1,     1, ...
  0,   2,    4,    2,     6,     4,     2,     2, ...
  0,   3,   12,    9,    27,    18,    15,     9, ...
  0,   8,   52,   32,   156,   100,    80,    56, ...
  0,  10,  240,  180,  1110,   650,   590,   360, ...
  0,  54, 1188,  954,  8322,  4908,  5034,  2934, ...
  0, -42, 6804, 6524, 70098, 41090, 47110, 26054, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=1-17, 37 give: A019590, A005727, A215524, A179230, A215704, A215522, A215705, A179405, A215706, A215707, A215708, A215709, A215691, A215710, A215643, A215629, A179505, A211205.
Rows n=0+1, 2-10 give: A000012, A215841, A215842, A215834, A215835, A215836, A215837, A215838, A215839, A215840.
Number of distinct values taken for m x's by derivatives n=1-10: A000012, A028310, A199085, A199205, A199296, A199883, A215796, A215971, A216062, A216403.
Main diagonal gives A306739.
Cf. A000081, A000108, A033917, A211192, A214569, A214570, A214571, A216041, A216281, A216349, A216350, A216351, A216368, A222379, A222380, A277537, A306679, A306710, A306726.

T:= proc(n) T(n):=`if`(n=1, [x], map(h-> x^h, g(n-1$2))) end:
g:= proc(n, i) option remember; `if`(i=1, [x^n], [seq(seq(
      seq(mul(T(i)[w[t]-t+1], t=1..j)*v, v=g(n-i*j, i-1)), w=
      combinat[choose]([$1..nops(T(i))+j-1], j)), j=0..n/i)])
    end:
f:= proc() local i, l; i, l:= 0, []; proc(n) while n>
      nops(l) do i:= i+1; l:= [l[], T(i)[]] od; l[n] end
    end():
A:= (n, k)-> n!*coeff(series(subs(x=x+1, f(k)), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=0..d), d=0..12);

T[n_] := If[n == 1, {x}, Map[x^#&, g[n - 1, n - 1]]];
g[n_, i_] := g[n, i] = If[i == 1, {x^n}, Flatten @ Table[ Table[ Table[ Product[T[i][[w[[t]] - t + 1]], {t, 1, j}]*v, {v, g[n - i*j, i - 1]}], {w, Subsets[ Range[ Length[T[i]] + j - 1], {j}]}], {j, 0, n/i}]];
f[n_] := Module[{i = 0, l = {}}, While[n > Length[l], i++; l = Join[l, T[i]]]; l[[n]]];
A[n_, k_] := n! * SeriesCoefficient[f[k] /. x -> x+1, {x, 0, n}];
Table[Table[A[n, 1+d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Nov 08 2019, after Alois P. Heinz *)

A052880 Expansion of e.g.f.: LambertW(1-exp(x))/(1-exp(x)).

Original entry on oeis.org

1, 1, 4, 26, 243, 2992, 45906, 845287, 18182926, 447797646, 12429760889, 384055045002, 13075708703910, 486430792977001, 19632714343389296, 854503410602781782, 39898063449977239323, 1989371798838577172796, 105503454201101084456182, 5930110732782743218645271

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple grammar.

Also the number of transitive reflexive early confluent binary relations R on n labeled elements. Early confluency means that (xRy and xRz) implies (yRz or zRy) for all x, y, z.

Alois P. Heinz, Table of n, a(n) for n = 0..378
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 851

Row sums of A135313.
Main diagonal of A135302.
Cf. A033917, A053763.

spec := [S,{B=Set(Z,1 <= card),S=Set(C),C=Prod(B,S)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
b:= proc(n, m) option remember; `if`(n=0,
     (m+1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 15 2022

CoefficientList[Series[-LambertW[-E^x+1]/(E^x-1), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
f[0, ] = 1; f[k, x_] := f[k, x] = Exp[Sum[x^m/m!*f[k-m, x], {m, 1, k}]];
(* b = A135302 *) b[0, 0] = 1; b[, 0] = 0; b[n, k_] := SeriesCoefficient[ f[k, x], {x, 0, n}]*n!;
a[n_] := b[n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 14 2019 *)

{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling2(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Mar 09 2013

x='x+O('x^30); Vec(serlaplace(-lambertw(-exp(x)+1)/(exp(x)-1))) \\ G. C. Greubel, Feb 19 2018

a(n) = Sum_{k=0..n} Stirling2(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ sqrt(1+exp(1)) * n^(n-1) / (exp(n-1)*(log(1+exp(1))-1)^(n-1/2)). - Vaclav Kotesovec, Nov 27 2012
E.g.f. A(x) satisfies: A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} Stirling2(n,k) * A(x)^k. - Paul D. Hanna, Mar 09 2013
E.g.f. A(x) satisfies: A(x) = exp((exp(x) - 1)*A(x)). - Ilya Gutkovskiy, Apr 04 2019

Edited by Alois P. Heinz, Nov 21 2010

A277537 A(n,k) is the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 2, 3, 0, 0, 1, 1, 2, 9, 8, 0, 0, 1, 1, 2, 9, 32, 10, 0, 0, 1, 1, 2, 9, 56, 180, 54, 0, 0, 1, 1, 2, 9, 56, 360, 954, -42, 0, 0, 1, 1, 2, 9, 56, 480, 2934, 6524, 944, 0, 0, 1, 1, 2, 9, 56, 480, 4374, 26054, 45016, -5112, 0, 0

Offset: 0

Alois P. Heinz, Oct 19 2016

			Square array A(n,k) begins:
  1, 1,   1,    1,     1,     1,     1,     1, ...
  0, 1,   1,    1,     1,     1,     1,     1, ...
  0, 0,   2,    2,     2,     2,     2,     2, ...
  0, 0,   3,    9,     9,     9,     9,     9, ...
  0, 0,   8,   32,    56,    56,    56,    56, ...
  0, 0,  10,  180,   360,   480,   480,   480, ...
  0, 0,  54,  954,  2934,  4374,  5094,  5094, ...
  0, 0, -42, 6524, 26054, 47894, 60494, 65534, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Columns k=0..10 give A000007, A019590(n+1), A005727, A179230, A179405, A179505, A211205, A277538, A277539, A277540, A277541.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A033917.
Cf. A215703, A277536, A295028.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
A:= (n, k)-> n!*coeff(series(f(k), x, n+1), x, n):
seq(seq(A(n, d-n), n=0..d), d=0..14);
# second Maple program:
b:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0,
      -add(binomial(n-1, j)*b(j, k)*add(binomial(n-j, i)*
      (-1)^i*b(n-j-i, k-1)*(i-1)!, i=1..n-j), j=0..n-1)))
    end:
A:= (n, k)-> b(n, min(k, n)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_] := b[n, k] = If[n==0, 1, If[k==0, 0, -Sum[Binomial[n-1, j]*b[j, k]*Sum[Binomial[n-j, i]*(-1)^i*b[n-j-i, k-1]*(i-1)!, {i, 1, n-j}], {j, 0, n-1}]]]; A[n_, k_] := b[n, Min[k, n]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 14 2017, adapted from 2nd Maple prog. *)

A(n,k) = [(d/dx)^n x^^k]_{x=1}.
E.g.f. of column k: (x+1)^^k.
A(n,k) = Sum_{i=0..min(n,k)} A277536(n,i).
A(n,k) = n * A295028(n,k) for n,k > 0.

A277522 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/2.

Original entry on oeis.org

5, 6, 9, 2, 4, 5, 2, 0, 4, 4, 2, 6, 3, 4, 8, 0, 6, 1, 0, 6, 5, 3, 3, 0, 4, 7, 7, 8, 4, 1, 9, 6, 6, 9, 0, 5, 2, 6, 3, 8, 6, 5, 9, 7, 3, 1, 4, 6, 3, 0, 2, 7, 6, 4, 5, 4, 9, 8, 0, 1, 1, 6, 7, 9, 1, 8, 3, 1, 1, 5, 4, 2, 1, 1, 3, 4, 6, 7, 0, 8, 7, 6, 2, 3, 2, 4, 0, 6, 7, 4, 8, 7, 9, 2, 6, 3, 5, 0, 2, 4, 5, 1, 5, 0, 3

Offset: 0

Alois P. Heinz, Oct 19 2016

			0.5692452044263480610653304778419669052638659731463...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277523, A277524, A277525, A277526, A277527, A277528, A277529, A277530, A277531, A277559, A277651, A293009, A300916.

RealDigits[2 Exp[-2 ProductLog[Log[2]]]/(1 + ProductLog[Log[2]]), 10, 105][[1]] (* Vladimir Reshetnikov, Oct 20 2016 *)
f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f'[1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

2*exp(-2*lambertw(log(2)))/(1+lambertw(log(2))) \\ G. C. Greubel, Nov 10 2017

Equals 2*exp(-2*LambertW(log(2)))/(1+LambertW(log(2))). - Vladimir Reshetnikov, Oct 20 2016

A277523 Decimal expansion of the second derivative of the infinite power tower function x^x^x... at x = 1/2.

Original entry on oeis.org

2, 2, 2, 1, 4, 0, 2, 1, 3, 6, 0, 1, 2, 2, 2, 1, 2, 6, 5, 5, 1, 5, 5, 3, 7, 3, 8, 5, 9, 6, 8, 0, 0, 3, 0, 8, 9, 5, 9, 1, 0, 8, 9, 7, 2, 6, 8, 6, 2, 8, 1, 5, 1, 7, 3, 8, 4, 7, 4, 4, 7, 7, 9, 8, 7, 0, 2, 1, 3, 9, 6, 9, 1, 7, 4, 7, 8, 5, 5, 1, 9, 0, 3, 9, 7, 5, 7, 2, 6, 5, 4, 2, 4, 2, 7, 1, 7, 8, 8, 4, 5, 2, 2, 5, 4

Offset: 0

Alois P. Heinz, Oct 19 2016

			0.222140213601222126551553738596800308959108972686281...

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748 (0th derivative), A277522, A277524, A277525, A277526, A277527, A277528, A277529, A277530, A277531.

RealDigits[4 LambertW[Log[2]]^2 ((2 - Log[2]) LambertW[Log[2]]^2 + (3 - 2 Log[2]) LambertW[Log[2]] - Log[2])/(Log[2] (1 + LambertW[Log[2]]))^3, 10, 105][[1]] (* Vladimir Reshetnikov, Oct 20 2016 *)
f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f''[1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

4*lambertw(log(2))^2*((2-log(2))*lambertw(log(2))^2 + (3-2*log(2)) *lambertw(log(2))-log(2))/(log(2)*(1+lambertw(log(2))))^3 \\ G. C. Greubel, Nov 10 2017

Equals 4 * LambertW(log(2))^2 * ((2-log(2)) * LambertW(log(2))^2 + (3-2*log(2)) * LambertW(log(2))-log(2)) / (log(2) * (1+LambertW(log(2))))^3. - Vladimir Reshetnikov, Oct 20 2016

A277524 Decimal expansion of the third derivative of the infinite power tower function x^x^x... at x = 1/2.

Original entry on oeis.org

1, 9, 0, 1, 7, 1, 3, 9, 0, 7, 5, 2, 6, 0, 0, 3, 9, 8, 6, 2, 6, 2, 6, 3, 9, 9, 8, 2, 1, 2, 4, 6, 9, 9, 5, 1, 4, 4, 4, 6, 4, 8, 8, 5, 7, 4, 5, 2, 4, 0, 5, 0, 5, 3, 1, 6, 2, 0, 0, 0, 9, 0, 0, 9, 1, 3, 5, 1, 5, 9, 4, 3, 5, 4, 5, 0, 7, 5, 5, 4, 2, 9, 0, 3, 4, 9, 2, 6, 8, 4, 0, 1, 4, 2, 1, 2, 6, 0, 2, 6, 6, 8, 4, 7, 1, 6, 5

Offset: 1

Alois P. Heinz, Oct 19 2016

			1.901713907526003986262639982124699514446488574524050...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277525, A277526, A277527, A277528, A277529, A277530, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[f'''[1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A277525 Decimal expansion of the fourth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

Original entry on oeis.org

1, 7, 2, 9, 6, 3, 7, 6, 7, 7, 6, 7, 5, 5, 4, 3, 4, 1, 4, 4, 7, 5, 9, 9, 4, 3, 6, 6, 8, 6, 0, 8, 3, 4, 0, 4, 1, 1, 3, 7, 8, 5, 9, 2, 4, 8, 5, 8, 7, 6, 2, 4, 7, 5, 4, 3, 6, 2, 8, 4, 5, 2, 7, 3, 6, 5, 9, 6, 9, 2, 3, 4, 8, 6, 2, 8, 9, 4, 3, 3, 4, 5, 2, 1, 7, 5, 6, 7, 1, 2, 3, 8, 4, 8, 3, 8, 7, 0, 4, 5, 2, 4, 4, 0, 8, 8

Offset: 1

Alois P. Heinz, Oct 19 2016

			-1.7296376776755434144759943668608340411378592485876...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277526, A277527, A277528, A277529, A277530, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[4][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A277526 Decimal expansion of the fifth derivative of the infinite power tower function x^x^x... at x = 1/2.

Original entry on oeis.org

4, 9, 2, 8, 5, 9, 0, 7, 6, 3, 7, 5, 8, 5, 1, 0, 1, 7, 7, 6, 0, 3, 3, 5, 1, 7, 8, 7, 8, 9, 2, 8, 5, 7, 1, 2, 5, 7, 9, 5, 7, 1, 4, 7, 8, 2, 5, 3, 8, 5, 0, 5, 4, 6, 8, 4, 6, 7, 2, 6, 6, 0, 0, 4, 7, 1, 2, 7, 6, 2, 0, 4, 1, 6, 3, 9, 6, 2, 1, 2, 9, 2, 1, 9, 5, 9, 8, 8, 5, 3, 1, 4, 8, 7, 9, 3, 7, 9, 1, 6, 2, 6, 4, 3, 0, 5, 7

Offset: 2

Alois P. Heinz, Oct 19 2016

			49.2859076375851017760335178789285712579571478...

Alois P. Heinz, Table of n, a(n) for n = 2..1001
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277525, A277527, A277528, A277529, A277530, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[5][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A277527 Decimal expansion of the sixth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

Original entry on oeis.org

2, 7, 7, 4, 4, 1, 3, 2, 0, 3, 5, 4, 0, 6, 1, 7, 9, 1, 0, 5, 8, 5, 6, 9, 5, 5, 1, 3, 9, 5, 9, 8, 4, 7, 5, 1, 4, 8, 8, 0, 3, 0, 9, 1, 5, 0, 5, 9, 4, 7, 0, 3, 9, 0, 8, 4, 1, 3, 5, 8, 3, 0, 8, 7, 8, 4, 7, 9, 7, 2, 0, 0, 0, 6, 0, 7, 2, 2, 5, 8, 9, 9, 7, 2, 5, 1, 5, 2, 7, 8, 1, 3, 3, 0, 5, 0, 1, 0, 8, 0, 0, 8, 8, 6, 0

Offset: 3

Alois P. Heinz, Oct 19 2016

			-277.441320354061791058569551395984751488030915...

Alois P. Heinz, Table of n, a(n) for n = 3..1002
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277525, A277526, A277528, A277529, A277530, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[6][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

A277528 Decimal expansion of the seventh derivative of the infinite power tower function x^x^x... at x = 1/2.

Original entry on oeis.org

4, 3, 4, 9, 0, 2, 8, 8, 5, 6, 7, 2, 8, 8, 3, 7, 8, 4, 8, 0, 0, 2, 2, 2, 5, 2, 1, 4, 0, 8, 3, 8, 9, 5, 1, 4, 9, 9, 1, 6, 9, 6, 1, 5, 3, 3, 9, 0, 0, 0, 4, 4, 3, 1, 4, 0, 1, 3, 9, 2, 2, 9, 5, 3, 7, 6, 1, 2, 6, 2, 8, 9, 6, 9, 8, 7, 1, 5, 7, 4, 4, 0, 8, 8, 5, 2, 1, 1, 6, 6, 7, 8, 1, 7, 3, 8, 1, 8, 5, 1, 7, 6, 7, 0, 9, 8, 5, 8

Offset: 4

Alois P. Heinz, Oct 19 2016

			4349.0288567288378480022252140838951499169615339...

Alois P. Heinz, Table of n, a(n) for n = 4..1003
Eric Weisstein's World of Mathematics, Power Tower.
Wikipedia, Tetration.

Cf. A033917, A104748, A277522, A277523, A277524, A277525, A277526, A277527, A277529, A277530, A277531.

f[x_] := -ProductLog[-Log[x]]/Log[x]; RealDigits[Derivative[7][f][1/2], 10, 120][[1]] (* Amiram Eldar, May 23 2023 *)

cp's OEIS Frontend

A215703 A(n,k) is the n-th derivative of f_k at x=1, and f_k is the k-th of all functions that are representable as x^x^...^x with m>=1 x's and parentheses inserted in all possible ways; square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A052880 Expansion of e.g.f.: LambertW(1-exp(x))/(1-exp(x)).

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A277537 A(n,k) is the n-th derivative of the k-th tetration of x (power tower of order k) x^^k at x=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A277522 Decimal expansion of the first derivative of the infinite power tower function x^x^x... at x = 1/2.

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A277523 Decimal expansion of the second derivative of the infinite power tower function x^x^x... at x = 1/2.

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A277524 Decimal expansion of the third derivative of the infinite power tower function x^x^x... at x = 1/2.

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Mathematica

A277525 Decimal expansion of the fourth derivative of the infinite power tower function x^x^x... at x = 1/2, negated.

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A277526 Decimal expansion of the fifth derivative of the infinite power tower function x^x^x... at x = 1/2.

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