A277537 -id:A277537 - 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 *)

A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.

Original entry on oeis.org

1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560, -61031188196889482880

Offset: 0

N. J. A. Sloane

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, table at foot of page.
G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.
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 (first 101 terms from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook), section 36.5, "The function x^x"
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y=xxy=x^x, Rocky Mountain J. Math. 26(2) 1996.
R. K. Guy, Letter to N. J. A. Sloane, 1986
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
G. H. Hardy, A Course of Pure Mathematics, Cambridge, The University Press, 1908.
D. H. Lehmer, Numbers associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, p. 461.
R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Row sums of A008296. Column k=2 of A215703 and of A277537.

Cf. A005168, A176118, A293297, A203852.

A005727 := proc(n) option remember; `if`(n=0, 1, A005727(n-1)+add((-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*A005727(k), k=0..n-2)) end:
seq(A005727(n), n=0..23); # Mélika Tebni, May 22 2022

NestList[ Factor[ D[ #1, x ] ]&, x^x, n ] /. (x->1)
Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)

a(n)=if(n<0,0,n!*polcoeff((1+x+x*O(x^n))^(1+x),n))

For n>0, a(n) = Sum_{k=0..n} b(n, k), where b(n, k) is a Lehmer-Comtet number of the first kind (see A008296).
E.g.f.: (1+x)^(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*A000248(k). - Vladeta Jovovic, Oct 02 2003
From  Mélika Tebni, May 22 2022: (Start)
a(0) = 1, a(n) = a(n-1)+Sum_{k=0..n-2} (-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*a(k).
a(n) = Sum_{k=0..n} (-1)^(n-k)*A293297(k)*binomial(n, k).
a(n) = Sum_{k=0..n} (-1)^k*A203852(k)*binomial(n, k). (End)

A033917 Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 65534, 984808, 16992144, 330667680, 7170714672, 171438170232, 4480972742064, 127115833240200, 3889913061111240, 127729720697035584, 4479821940873927168, 167143865005981109952, 6610411989494027218368, 276242547290322145178880

Offset: 0

Patrick D McLean

a(n) is the n-th derivative of x^(x^...(x^(x^x))) with n x's evaluated at x=1. - Alois P. Heinz, Oct 14 2016

Alois P. Heinz, Table of n, a(n) for n = 0..400
R. Arthur Knoebel, Exponentials Reiterated, Amer. Math. Monthly 88 (1981), pp. 235-252.
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Cf. A052880, A215703, A216349, A216350.
Row sums of A277536.
Main diagonal of A277537.

a:= n-> add(Stirling1(n, k)*(k+1)^(k-1), k=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 31 2012

mx = 20; Table[ i! SeriesCoefficient[ InverseSeries[ Series[ y^(1/y), {y, 1, mx}]], i], {i, 0, n}] (* modified by Robert G. Wilson v, Feb 03 2013 *)
CoefficientList[Series[-LambertW[-Log[1+x]]/Log[1+x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)

Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)
a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, sum(k=0, m, Stirling1(m, k)*(A+x*O(x^n))^k)*x^m/m!)); n!*polcoeff(A, n)
for(n=0,20,print1(a(n),", ")) \\ Paul D. Hanna, Mar 09 2013

E.g.f.: -LambertW(-log(1+x))/log(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*(k+1)^(k-1). - Vladeta Jovovic, Nov 12 2003
a(n) ~ n^(n-1) / ( exp(n -3/2 + exp(-1)/2) * (exp(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} Stirling1(n,k) * A(x)^k. - Paul D. Hanna, Mar 09 2013
a(n) = n! * [x^n] (x+1)^^n. - Alois P. Heinz, Oct 19 2016

A179230 n-th derivative of x^(x^x) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 32, 180, 954, 6524, 45016, 360144, 3023640, 27617832, 271481880, 2775329232, 31188079272, 350827041000, 4441125248640, 54110311240512, 765546040603584, 9938498593229568, 156934910753107200, 2128783325724881280, 37775147271084647424

Offset: 0

Henryk Trappmann (bo198214(AT)gmail.com), Jul 03 2010

sign

First term < 0: a(33) = -868875490363254484795699722301440.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727. Column k=4 of A215703. Column k=3 of A277537.

Cf. A295103.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^x) ), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 20 2012

Table[ D[ x^(x^x), {x, n}] /. x -> 1, {n, 0, 20}] (* Robert G. Wilson v, Jul 12 2010 *)
NestList[ Factor[ D[ #1, x]] &, x^x^x, 20] /. x -> 1 (* Robert G. Wilson v, Aug 10 2010 *)
Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^(x+1)). - Alois P. Heinz, Aug 21 2012

a(21)-a(22) from Alois P. Heinz, Aug 20 2012

A179405 n-th derivative of x^(x^(x^x)) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 360, 2934, 26054, 269128, 3010680, 37616880, 504880992, 7387701672, 115228447152, 1929016301016, 34194883090440, 643667407174464, 12757366498618176, 266426229010029696, 5830527979298793024, 133665090871032478080, 3197905600674249843840

Offset: 0

Robert G. Wilson v, Jul 13 2010

sign

First term < 0: a(329). - Alois P. Heinz, Sep 22 2015

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727, A179230, A295104.
Column k=8 of A215703.
Column k=4 of A277537.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^x)) ), x, n+1), x, n):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 20 2012

f[n_] := D[ x^(x^(x^x)), {x, n}] /. x -> 1; Array[f, 18, 0]
Range[0, 21]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 21}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^((x+1)^(x+1))). - Alois P. Heinz, Aug 23 2012

a(18)-a(21) from Alois P. Heinz, Aug 20 2012

A295028 A(n,k) is (1/n) times 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>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 3, 2, 0, 1, 1, 3, 8, 2, 0, 1, 1, 3, 14, 36, 9, 0, 1, 1, 3, 14, 72, 159, -6, 0, 1, 1, 3, 14, 96, 489, 932, 118, 0, 1, 1, 3, 14, 96, 729, 3722, 5627, -568, 0, 1, 1, 3, 14, 96, 849, 6842, 33641, 40016, 4716, 0

Offset: 1

Alois P. Heinz, Nov 12 2017

			Square array A(n,k) begins:
  1,   1,    1,     1,     1,      1,      1,      1, ...
  0,   1,    1,     1,     1,      1,      1,      1, ...
  0,   1,    3,     3,     3,      3,      3,      3, ...
  0,   2,    8,    14,    14,     14,     14,     14, ...
  0,   2,   36,    72,    96,     96,     96,     96, ...
  0,   9,  159,   489,   729,    849,    849,    849, ...
  0,  -6,  932,  3722,  6842,   8642,   9362,   9362, ...
  0, 118, 5627, 33641, 71861, 102941, 118061, 123101, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Columns k=1-10 give: A063524, A005168, A295103, A295104, A295105, A295106, A295107, A295108, A295109, A295110.
Main diagonal gives A136461(n-1).
Cf. A277537, A295027.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
A:= (n, k)-> (n-1)!*coeff(series(f(k), x, n+1), x, n):
seq(seq(A(n, 1+d-n), n=1..d), d=1..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))/n:
seq(seq(A(n, 1+d-n), n=1..d), d=1..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]]/n;
Table[A[n, 1 + d - n], {d, 1, 14}, {n, 1, d}] // Flatten (* Jean-François Alcover, May 25 2018, translated from 2nd Maple program *)

A(n,k) = 1/n * [(d/dx)^n x^^k]_{x=1}.
A(n,k) = (n-1)! * [x^n] (x+1)^^k.
A(n,k) = Sum_{i=0..min(n,k)} A295027(n,i).
A(n,k) = 1/n * A277537(n,k).

A179505 n-th derivative of x^(x^(x^(x^x))) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 4374, 47894, 574888, 7829424, 116392080, 1901059512, 33564909432, 639562529424, 13047133134840, 283976169754440, 6563364026374464, 160538113862231808, 4141949353327046592, 112396373034208003008, 3199752121483607518080

Offset: 0

Robert G. Wilson v, Jul 17 2010

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727, A179230, A179405, A295105.
Column k=17 of A215703.
Column k=5 of A277537.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^(x^x))) ), x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 21 2012

f[n_] := D[x^(x^(x^(x^x))), {x, n}] /. x -> 1; Array[f, 16, 0]
Range[0, 20]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 20}], x] (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: (x+1)^((x+1)^((x+1)^((x+1)^(x+1)))). - Alois P. Heinz, Aug 21 2012

a(16)-a(20) from Alois P. Heinz, Aug 21 2012

A277536 T(n,k) is the n-th derivative of the difference between the k-th tetration of x (power tower of order k) and its predecessor (or 0 if k=0) at x=1; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 3, 6, 0, 0, 8, 24, 24, 0, 0, 10, 170, 180, 120, 0, 0, 54, 900, 1980, 1440, 720, 0, 0, -42, 6566, 19530, 21840, 12600, 5040, 0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320, 0, 0, -5112, 365256, 2650536, 4818744, 4536000, 2993760, 1270080, 362880

Offset: 0

Alois P. Heinz, Oct 19 2016

T(n,k) is defined for all n,k >= 0. The triangle contains only the terms with k<=n. T(n,k) = 0 for k>n.

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0,   2;
  0, 0,   3,     6;
  0, 0,   8,    24,     24;
  0, 0,  10,   170,    180,    120;
  0, 0,  54,   900,   1980,   1440,    720;
  0, 0, -42,  6566,  19530,  21840,  12600,   5040;
  0, 0, 944, 44072, 224112, 305760, 248640, 120960, 40320;
  ...

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

Columns k=0-2 give: A000007, A063524, A005727 (for n>1).
Main diagonal gives A000142.
Row sums give A033917.
T(n+1,n)/3 gives A005990.
T(2n,n) gives A290023.
Cf. A277537, A295027.

f:= proc(n) option remember; `if`(n<0, 0,
      `if`(n=0, 1, (x+1)^f(n-1)))
    end:
T:= (n, k)-> n!*coeff(series(f(k)-f(k-1), x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..12);
# 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:
T:= (n, k)-> b(n, min(k, n))-`if`(k=0, 0, b(n, min(k-1, n))):
seq(seq(T(n, k), k=0..n), n=0..12);

f[n_] := f[n] = If[n < 0, 0, If[n == 0, 1, (x + 1)^f[n - 1]]];
T[n_, k_] := n!*SeriesCoefficient[f[k] - f[k - 1], { x, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
(* second program: *)
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}]]];
T[n_, k_] := (b[n, Min[k, n]] - If[k == 0, 0, b[n, Min[k - 1, n]]]);
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 28 2018, from Maple *)

E.g.f. of column k>0: (x+1)^^k - (x+1)^^(k-1), e.g.f. of column k=0: 1.
T(n,k) = [(d/dx)^n (x^^k - x^^(k-1))]_{x=1} for k>0, T(n,0) = A000007(n).
T(n,k) = A277537(n,k) - A277537(n,k-1) for k>0, T(n,0) = A000007(n).
T(n,k) = n * A295027(n,k) for n,k > 0.

A211205 n-th derivative of x^(x^(x^(x^(x^x)))) at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 60494, 823528, 12365424, 206078880, 3745686912, 74083090872, 1579529362944, 36165466533000, 884104045301640, 22992315801392064, 633547543117707648, 18439576158792912192, 565162707747635408448, 18194047307015185486080

Offset: 0

Alois P. Heinz and Robert G. Wilson v, Feb 04 2013

nonn

Robert G. Wilson v and Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A005727, A008405, A176118, A179230, A179405, A179505, A215522, A215524, A215629, A215643, A215691, A215704, A215705, A215706, A215707, A215708, A215709, A215710, A215522, A295106.
Column k=37 of A215703.
Column k=6 of A277537.

a:= n-> n!*coeff(series(subs(x=x+1, x^(x^(x^(x^(x^x))))), x, n+1), x, n):
seq(a(n), n=0..20);

NestList[ Factor[ D[#1, x]] &, x^x^x^x^x^x, 9] /. (x -> 1) (* or quicker *)
Range[0, 20]! CoefficientList[ Series[(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x)^(1 + x), {x, 0, 20}], x]

E.g.f.: (x+1)^((x+1)^((x+1)^((x+1)^((x+1)^(x+1))))).

A277538 n-th derivative of the seventh tetration of x (power tower of order 7) x^^7 at x=1.

Original entry on oeis.org

1, 1, 2, 9, 56, 480, 5094, 65534, 944488, 15359184, 274118880, 5369469072, 114055774392, 2618129199504, 64505645760360, 1699067695202040, 47625773113856064, 1415693345589370368, 44474719907550007872, 1472352462644660486208, 51227002322058534554880

Offset: 0

Alois P. Heinz, Oct 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
Eric Weisstein's World of Mathematics, Power Tower
Wikipedia, Knuth's up-arrow notation
Wikipedia, Tetration

Column k=7 of A277537.

Cf. A215703, A295107.

f:= proc(n) f(n):= `if`(n=0, 1, (x+1)^f(n-1)) end:
a:= n-> n!*coeff(series(f(7), x, n+1), x, n):
seq(a(n), n=0..25);

f[0] = 1; f[n_] := f[n] = (x + 1)^f[n - 1];
a[n_] := n! SeriesCoefficient[f[7], {x, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 29 2018, from Maple *)

E.g.f.: (x+1)^^7.

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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A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.

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A033917 Coefficients of iterated exponential function defined by y(x) = x^y(x) for e^-e < x < e^(1/e), expanded about x=1.

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A179230 n-th derivative of x^(x^x) at x=1.

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A179405 n-th derivative of x^(x^(x^x)) at x=1.

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A295028 A(n,k) is (1/n) times 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>=1, k>=1, read by antidiagonals.

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A179505 n-th derivative of x^(x^(x^(x^x))) at x=1.

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