A215703 -id:A215703 - cp's OEIS frontend

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

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

A215524 n-th derivative of (x^x)^x at x=1.

Original entry on oeis.org

1, 1, 4, 12, 52, 240, 1188, 6804, 38960, 253296, 1654560, 11816640, 85816608, 668005728, 5240582592, 44667645120, 365989405440, 3494595006720, 28075694694912, 325862541872640, 2101211758356480, 39605981661066240, 48568198208302080, 7549838510211486720, -66667098077331572736

Offset: 0

Alois P. Heinz, Aug 14 2012

sign

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

Column k=3 of A215703.

Cf. A005727, A179230, A179405, A215522.

a:= n-> n!*coeff(series(subs(x=x+1, (x^x)^x ), x, n+1), x, n):
seq(a(n), n=0..30);

m = 24; CoefficientList[((x+1)^(x+1))^(x+1) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

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

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.

A199085 Number of distinct values taken by 3rd derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 15, 20, 26, 32, 39, 47, 55, 64, 74, 84, 95, 107, 119, 132, 146, 160, 175, 191, 207, 224, 242, 260, 279, 299, 319, 340, 362, 384, 407, 431, 455, 480, 506, 532, 559, 587, 615, 644, 674, 704, 735, 767, 799, 832, 866, 900, 935, 971, 1007, 1044

Offset: 1

Vladimir Reshetnikov, Nov 02 2011

nonn

Number of distinct values taken by 0th and 1st derivative is 1,1,1,1,1,1,... and by 2nd is 1,1,2,3,4,5,...

			For n=5 there are 7 distinct values: 9, 15, 18, 21, 24, 33, 48, they are given by 3rd derivatives of the following parenthesizations at x=1: x^(x^((x^x)^x)), x^((x^(x^x))^x), (x^x)^(x^(x^x)), x^(((x^x)^x)^x), (x^(x^x))^(x^x), (((x^(x^x)))^x)^x, (((x^x)^x)^x)^x. So a(5)=7.

Cf. A000081 (upper bound), A000108, A199205 (4th derivatives), A199296 (5th derivatives), A215703, A215842. Column k=3 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[0, 0]}, {seq(seq(seq([2+g[1], 3*(1+g[1]+h[1])+g[2]],
                           h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[2], f(n))):
seq(a(n), n=1..40);  # Alois P. Heinz, Nov 03 2011

f[1] = {x}; f[n_] := Flatten[Table[Outer[Power, f[k], f[n-k]], {k, n-1}]]; Table[Length[Union[D[f[n], {x, 3}] /. x -> 1]], {n, 1, 8}] (* Reshetnikov *)
Table[If[n<3,1,Floor[(n^2-2)/3]],{n,1,100}] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2012 *)

Conjectured g.f.: (x^6-2*x^4-x^2+x-1)*x/(x^5-2*x^4+x^3-x^2+2*x-1). - Alois P. Heinz, Nov 02 2011

a(13)-a(56) from Alois P. Heinz, Nov 02 2011

A199205 Number of distinct values taken by 4th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 17, 30, 50, 77, 113, 156, 212, 279, 355, 447, 560, 684, 822, 985, 1171, 1375, 1599, 1856, 2134, 2445, 2769, 3125, 3519, 3939, 4376, 4857, 5372, 5914, 6484, 7083, 7717, 8411, 9130, 9882, 10683, 11524, 12393

Offset: 1

Alois P. Heinz, Nov 03 2011

nonn

			a(5) = 9 because the A000108(4) = 14 possible parenthesizations of x^x^x^x^x lead to 9 different values of the 4th derivative at x=1: (x^(x^(x^(x^x)))) -> 56; (x^(x^((x^x)^x))) -> 80; (x^((x^(x^x))^x)), (x^((x^x)^(x^x))) -> 104; ((x^x)^(x^(x^x))), ((x^(x^(x^x)))^x) -> 124; ((x^(x^x))^(x^x)) -> 148; (x^(((x^x)^x)^x)) -> 152; ((x^x)^((x^x)^x)), ((x^((x^x)^x))^x) -> 172; (((x^x)^x)^(x^x)), (((x^(x^x))^x)^x), (((x^x)^(x^x))^x) -> 228; ((((x^x)^x)^x)^x) -> 344.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199296 (5th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215834. Column k=4 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0]},
                {seq(seq(seq( [2+g[1], 3*(1 +g[1] +h[1]) +g[2],
                 8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3]],
                 h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[3], f(n))):
seq(a(n), n=1..20);

f[n_] := f[n] = If[n == 1, {{0, 0, 0}}, Union @ Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length @ Union @ (#[[3]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 32}] (* Jean-François Alcover, Jun 08 2018, after Alois P. Heinz *)

a(41)-a(42) from Alois P. Heinz, Jun 01 2015

A199296 Number of distinct values taken by 5th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 45, 92, 182, 342, 601, 982, 1499, 2169, 2970, 3994, 5297, 6834, 8635, 10714, 13121, 16104, 19674, 23868, 28453, 33637, 39630, 46730

Offset: 1

Alois P. Heinz, Nov 04 2011

nonn

			a(4) = 4 because the A000108(3) = 5 possible parenthesizations of x^x^x^x lead to 4 different values of the 5th derivative at x=1: (x^(x^(x^x))) -> 360; (x^((x^x)^x)) -> 590; ((x^(x^x))^x), ((x^x)^(x^x)) -> 650; (((x^x)^x)^x) -> 1110.

Cf. A000081 (distinct functions), A000108 (parenthesizations), A000012 (first derivatives), A028310 (2nd derivatives), A199085 (3rd derivatives), A199205 (4th derivatives), A002845, A003018, A003019, A145545, A145546, A145547, A145548, A145549, A145550, A082499, A196244, A198683, A215703, A215835. Column k=5 of A216368.

f:= proc(n) option remember;
      `if`(n=1, {[0, 0, 0, 0]},
            {seq(seq(seq([2+g[1], 3*(1 +g[1] +h[1]) +g[2],
             8 +12*g[1] +6*h[1]*(1+g[1]) +4*(g[2]+h[2])+g[3],
             10+50*h[1]+10*h[2]+5*h[3]+(30+30*h[1]+10*h[2]
             +15*g[1])*g[1]+(20+10*h[1])*g[2]+5*g[3]+g[4]],
              h=f(n-j)), g=f(j)), j=1..n-1)})
    end:
a:= n-> nops(map(x-> x[4], f(n))):
seq(a(n), n=1..20);

f[n_] := f[n] = If[n == 1, {{0, 0, 0, 0}}, Union@Flatten[#, 3]& @ {Table[ Table[Table[{2 + g[[1]], 3*(1 + g[[1]] + h[[1]]) + g[[2]], 8 + 12*g[[1]] + 6*h[[1]]*(1 + g[[1]]) + 4*(g[[2]] + h[[2]]) + g[[3]], 10 + 50*h[[1]] + 10*h[[2]] + 5*h[[3]] + (30 + 30*h[[1]] + 10*h[[2]] + 15*g[[1]])*g[[1]] + (20 + 10*h[[1]])*g[[2]] + 5*g[[3]] + g[[4]]}, {h, f[n - j]}], {g, f[j]}], {j, 1, n - 1}]}];
a[n_] := Length@Union@(#[[4]]& /@ f[n]);
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* Jean-François Alcover, Sep 01 2023, after Alois P. Heinz *)

A216368 Number T(n,k) of distinct values taken by k-th derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 7, 9, 9, 1, 5, 11, 17, 20, 20, 1, 6, 15, 30, 45, 48, 48, 1, 7, 20, 50, 92, 113, 115, 115, 1, 8, 26, 77, 182, 262, 283, 286, 286, 1, 9, 32, 113, 342, 591, 691, 717, 719, 719, 1, 10, 39, 156, 601, 1263, 1681, 1815, 1838, 1842, 1842

Offset: 1

Alois P. Heinz, Sep 05 2012

T(n,k) <= A000081(n) because there are only A000081(n) different functions that can be represented with n x's.
It is not true that T(n,n) = T(n,n-1) for all n>1: T(13,13) - T(13,12) = 12486 - 12485 = 1.
Conjecture: T(n,n) = A000081(n) for n>=1. It would be nice to have a proof (or a disproof if the conjecture is wrong).
From Bradley Klee, Jun 01 2015 (Start):
I made a descendant graph (Plot 1) that shows how each derivative relates to the next. In this picture the number of nodes in row k gives the value T(n,k).  You can see at n=6 collisions begin to occur, and at n=7 the situation is even worse.  I then computed a new triangle with collisions removed (Plot 2) and values:
    1
    1 1
    1 2  2
    1 3  4  4
    1 4  7  9  9
    1 5 11 88 20 20
    1 6 16 34 46 48 48
I suspect that Plot 2 will admit a recursive construction more readily than the graphs with collisions. You can already see that each graph "n-1" is a subgraph of graph "n" and that the remainder of graph "n" is similar to graph "n-1" with additional branches. (End)

			For n = 4 there are A000108(3) = 5 possible parenthesizations of x^x^x^x: [x^(x^(x^x)), x^((x^x)^x), (x^(x^x))^x, (x^x)^(x^x), ((x^x)^x)^x]. The first, second, third, fourth derivatives at x=1 are [1,1,1,1,1], [2,2,4,4,6], [9,15,18,18,27], [56,80,100,100,156] => row 4 = [1,3,4,4].
Triangle T(n,k) begins:
  1;
  1, 1;
  1, 2,  2;
  1, 3,  4,  4;
  1, 4,  7,  9,  9;
  1, 5, 11, 17, 20,  20;
  1, 6, 15, 30, 45,  48,  48;
  1, 7, 20, 50, 92, 113, 115, 115;
  ...

Alois P. Heinz, Rows n = 1..16, flattened
Bradley Klee, Plot 1
Bradley Klee, Plot 2

Columns k=1-10 give: A000012, A028310, A199085, A199205, A199296, A199883, A215796, A215971, A216062, A216403.
Main diagonal gives (conjectured): A000081.
Cf. A000108, A215703.

with(combinat):
F:= proc(n) F(n):=`if`(n<2, [(x+1)$n], map(h->(x+1)^h, g(n-1, n-1))) end:
g:= proc(n, i) option remember; `if`(n=0 or i=1, [(x+1)^n],
     `if`(i<1, [], [seq(seq(seq(mul(F(i)[w[t]-t+1], t=1..j)*v,
      w=choose([$1..nops(F(i))+j-1], j)), v=g(n-i*j, i-1)), j=0..n/i)]))
    end:
T:= proc(n) local i, l;
      l:= map(f->[seq(i!*coeff(series(f, x, n+1), x, i), i=1..n)], F(n));
      seq(nops({map(x->x[i], l)[]}), i=1..n)
    end:
seq(T(n), n=1..10);

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}]];
T[n_] := T[n] = If[n==1, {x}, x^#& /@ g[n-1, n-1]];
T[n_, k_] := Union[k! (SeriesCoefficient[#, {x, 0, k}]& /@ (T[n] /. x -> x+1))] // Length;
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

cp's OEIS Frontend

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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A215524 n-th derivative of (x^x)^x at x=1.

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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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A199085 Number of distinct values taken by 3rd derivative of x^x^...^x (with n x's and parentheses inserted in all possible ways) at x=1.

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