A005727 -id:A005727 - cp's OEIS frontend

A349561 E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).

1, 1, 0, 3, -8, 100, -834, 11438, -159928, 2762352, -52322160, 1124320032, -26509832040, 686751503568, -19306448087640, 586539826169880, -19131996548499264, 667157522614934016, -24762890955027112128, 974824890777753840576, -40566428716555791936000

Offset: 0

Seiichi Manyama, Nov 22 2021

sign

			A(x) - 1 = x + 3*x^3/6 - 8*x^4/24 + ... = x + x^3/2 - x^4/3 + ... .
A(x)^A(x) = (1 + (A(x) - 1))^(1 + (A(x) - 1)) = Sum_{k>=0} A005727(k) * (A(x) - 1)^k / k! = 1 + 1 * (x + x^3/2 - x^4/3 + ... )/1! + 2 * (x + x^3/2 - x^4/3 + ... )^2/2! + 3 * (x + x^3/2 - x^4/3 + ... )^3/3! + ...  = 1 + x + x^2 + x^3 + ... = 1/(1 - x).

Seiichi Manyama, Table of n, a(n) for n = 0..398
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A005727, A120980, A141209, A216135, A216136, A229237, A305819.

Join[{1}, Table[(n-1)! - (-1)^n*Sum[(k-1)^(k-1)*StirlingS1[n, k], {k, 2, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Nov 22 2021 *)

a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 1));

my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (k-1)^(k-1)*log(1-x)^k/k!)))

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

a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling1(n,k).
E.g.f. A(x) = -Sum_{k>=0} (k-1)^(k-1) * (log(1-x))^k / k!.
E.g.f.: A(x) = -log(1-x)/LambertW(-log(1-x)).
a(n) ~ -(-1)^n * n^(n-1) / ((exp(exp(-1)) - 1)^(n - 1/2) * exp(n + exp(-1)/2 + 1/2)). - Vaclav Kotesovec, Nov 22 2021

A176118 The n-th derivative of 1/x^x, evaluated at x=1.

Original entry on oeis.org

1, -1, 0, 3, -8, 10, 6, -42, -160, 2952, -27720, 253440, -2553528, 28562664, -349272000, 4618376280, -65615072640, 996952226880, -16133983959744, 277093189849536, -5033937521116800, 96451913892983040, -1943937259314019200, 41112770486238380160

Offset: 0

Jacob Parr (jacobparr1(AT)gmail.com), Apr 09 2010

sign

			E.g.f.: A(x) = 1 - x + 3*x^3/3! - 8*x^4/4! + 10*x^5/5! + 6*x^6/6! - 42*x^7/7! - 160*x^8/8! + 2952*x^9/9! - 27720*x^10/10! + 253440*x^11/11! + ...
The e.g.f. as a power series with reduced fractional coefficients begins
A(x) = 1 - x + 1/2x^3 - 1/3x^4 + 1/12x^5 + 1/120x^6 - 1/120x^7 - 1/252x^8 + 41/5040x^9 - 11/1440x^10 + 2/315x^11 - 106397/19958400x^12 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A005727, A008296, A194786.

1, seq(simplify(subs(x = 1, diff(x^(-x), `$`(x, n)))), n = 1 .. 22); # Emeric Deutsch, Apr 14 2010
a:= n-> n! *coeftayl(x^(-x), x=1, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 18 2012

NestList[Factor[D[#1, x]] &, 1/x^x, 22] /. (x -> 1) (* Robert G. Wilson v, Feb 03 2013 *)

E.g.f.: 1 + x*(Q(0) - 1)/(x+1) where Q(k) = 1 - (1+x/(k+1))/(1 - x/(x + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 05 2013
a(n) ~ (-1)^(n+1) * n! / n^2. - Vaclav Kotesovec, Sep 03 2014
E.g.f.: 1/(x+1)^(x+1). - Alois P. Heinz, Sep 27 2016
a(n) = Sum_{k=0..n} (-1)^k * A008296(n,k). - Alois P. Heinz, Aug 25 2021
E.g.f.: Sum_{n>=0} (-1)^n * x^n/n! * Product_{k=1..n} (k + x). - Paul D. Hanna, Nov 13 2023

Definition edited by Emeric Deutsch, Apr 14 2010

More terms from Emeric Deutsch and R. J. Mathar, Apr 14 2010

A215522 n-th derivative of (x^x)^(x^x) at x=1.

Original entry on oeis.org

1, 1, 4, 18, 100, 650, 4908, 41090, 382520, 3863808, 42409440, 497972112, 6259762320, 83343114504, 1175904241848, 17442325040520, 272149555445760, 4438451554802880, 75714874759039104, 1343817666163911168, 24837691533530152320, 475811860099666527360

Offset: 0

Alois P. Heinz, Aug 14 2012

sign

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

First term < 0: a(65).

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

Column k=6 of A215703.

Cf. A005727, A179230, A179405, A215524.

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

m = 21; CoefficientList[(x+1)^((x+1)^(x+2)) + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, Feb 07 2021 *)

E.g.f.: (x+1)^((x+1)^(x+2)).

A007120 Expansion of e.g.f. (1+x)^(1-x).

Original entry on oeis.org

1, 1, -2, -3, 16, -10, -114, 462, -496, -2952, 16920, -31680, -130008, 1707576, -14259504, 138375720, -1652311680, 22238105280, -321916019904, 4959460972224, -81123831838080, 1405677742882560, -25732182778727040, 496324987642915200, -10061966722201900416

Offset: 0

Simon Plouffe

sign

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..453

Cf. A005727.

a(n) = n!*polcoef((1+x+x*O(x^n))^(1-x), n); \\ Seiichi Manyama, Sep 13 2021

Signs from Christian G. Bower, Nov 15 1998

Signs corrected by Sean A. Irvine, Oct 17 2017

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.

A354795 Triangle read by rows. The matrix inverse of A354794. Equivalently, the Bell transform of cfact(n) = -(n - 1)! if n > 0 and otherwise 1/(-n)!.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -3, 1, 0, -2, -1, -6, 1, 0, -6, 0, 5, -10, 1, 0, -24, 4, 15, 25, -15, 1, 0, -120, 28, 49, 35, 70, -21, 1, 0, -720, 188, 196, 49, 0, 154, -28, 1, 0, -5040, 1368, 944, 0, -231, -252, 294, -36, 1, 0, -40320, 11016, 5340, -820, -1365, -987, -1050, 510, -45, 1

Offset: 0

Peter Luschny, Jun 09 2022

The triangle is the matrix inverse of the Bell transform of n^n (A354794).
The numbers (-1)^(n-k)*T(n, k) are known as the Lehmer-Comtet numbers of 1st kind (A008296).
The function cfact is the 'complementary factorial' (name is ad hoc) and written \hat{!} in TeX mathmode. 1/(cfact(-n) * cfact(n)) = signum(-n) * n for n != 0. It is related to the Roman factorial (A159333). The Bell transform of the factorial are the Stirling cycle numbers (A132393).

			Triangle T(n, k) begins:
[0] [1]
[1] [0,     1]
[2] [0,    -1,    1]
[3] [0,    -1,   -3,   1]
[4] [0,    -2,   -1,  -6,   1]
[5] [0,    -6,    0,   5, -10,    1]
[6] [0,   -24,    4,  15,  25,  -15,    1]
[7] [0,  -120,   28,  49,  35,   70,  -21,   1]
[8] [0,  -720,  188, 196,  49,    0,  154, -28,   1]
[9] [0, -5040, 1368, 944,   0, -231, -252, 294, -36, 1]

Louis Comtet, Advanced Combinatorics. Reidel, Dordrecht, 1974, p. 139-140.

D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.
Peter Luschny, The Bell transform

Cf. A354794 (matrix inverse), A176118 (row sums), A005727 (alternating row sums), A045406 (column 2), A347276 (column 3), A345651 (column 4), A298511 (central), A008296 (variant), A159333, A264428, A159075, A006963, A354796.

# The function BellMatrix is defined in A264428.
cfact := n -> ifelse(n = 0, 1, -(n - 1)!): BellMatrix(cfact, 10);
# Alternative:
t := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = n then 1 else (n-1)*t(n-2, k-1) - (n-1-k)*t(n-1, k) + t(n-1, k-1) fi end:
T := (n, k) -> (-1)^(n-k)*t(n, k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..9);
# Using the e.g.f.:
egf := (1 - x)^(t*(x - 1)):
ser := series(egf, x, 11): coeffx := n -> coeff(ser, x, n):
row := n -> seq(n!*coeff(coeffx(n), t, k), k=0..n):
seq(print(row(n)), n = 0..9);

cfact[n_] := If[n == 0, 1, -(n - 1)!];
R := Range[0, 10]; cf := Table[cfact[n], {n, R}];
Table[BellY[n, k, cf], {n, R}, {k, 0, n}] // Flatten

T(n, k) = n!*[t^k][x^n] (1 - x)^(t*(x - 1)).
T(n, k) = Sum_{j=k..n} (-1)^(n-k)*binomial(j, k)*k^(j-k)*Stirling1(n, j).
T(n, k) = Bell_{n, k}(a), where Bell_{n, k} is the partial Bell polynomial evaluated over the sequence a = {cfact(m) | m >= 0}, (see Mathematica).
T(n, k) = (-1)^(n-k)*t(n, k) where t(n, n) = 1 and t(n, k) = (n-1)*t(n-2, k-1) - (n-1-k)*t(n-1, k) + t(n-1, k-1) for k > 0 and n > 0.
Let s(n) = (-1)^n*Sum_{k=1..n} (k-1)^(k-1)*T(n, k) for n >= 0, then s = A159075.
Sum_{k=1..n} (k + x)^(k-1)*T(n, k) = binomial(n + x - 1, n-1)*(n-1)! for n >= 1. Note that for x = k this is A354796(n, k) for 0 <= k <= n and implies in particular for x = n >= 1 the identity Sum_{k=1..n} (k + n)^(k - 1)*T(n, k) = Gamma(2*n)/n! = A006963(n+1).
E.g.f. of column k >= 0: ((1 - t) * log(1 - t))^k / ((-1)^k * k!). - Werner Schulte, Jun 14 2022

A215704 n-th derivative of ((x^x)^x)^x at x=1.

Original entry on oeis.org

1, 1, 6, 27, 156, 1110, 8322, 70098, 646272, 6333336, 66712680, 745731360, 8780828328, 108873486072, 1413807287760, 19157627737080, 270460073295360, 3965693824244160, 60266513065134528, 947644484349584448, 15389579447794454400, 257702782790624613120

Offset: 0

Alois P. Heinz, Aug 21 2012

sign

Also n-th derivative of x^(x^3) at x=1.

First term < 0: a(57).

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

Column k=5 of A215703.

Cf. A005727, A179230, A179405, A215524.

a:= n-> n!*coeff(series(subs(x=x+1, ((x^x)^x)^x ), x, n+1), x, n):
seq(a(n), n=0..25);

E.g.f.: (x+1)^((x+1)^3).

A005168 n-th derivative of x^x at 1, divided by n.

Original entry on oeis.org

1, 1, 1, 2, 2, 9, -6, 118, -568, 4716, -38160, 358126, -3662088, 41073096, -500013528, 6573808200, -92840971200, 1402148010528, -22554146644416, 385014881294496, -6952611764874240, 132427188835260480, -2653529921603890560, 55802195178451990896

Offset: 1

N. J. A. Sloane, R. K. Guy

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 = 1..400 (first 100 terms from T. D. Noe)
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]
R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Cf. A005727.

Column k=2 of A295027 (for n>1), A295028.

a:= n-> (n-1)! *coeftayl(x^x, x=1, n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 18 2012

Rest[(NestList[ Factor[ D[ #1, x]] &, x^x, 23] /. (x -> 1))/Range[0, 23]] (* Robert G. Wilson v, Aug 10 2010 *)

from sympy import var, diff
x = var('x')
y = x**x
l = [[y:=diff(y),y.subs(x,1)/(n+1)][1] for n in range(10)]
print(l) # Nicholas Stefan Georgescu, Mar 02 2023

One more term from Robert G. Wilson v, Aug 10 2010

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))))).

cp's OEIS Frontend

A349561 E.g.f. satisfies: A(x)^A(x) = 1/(1 - x).

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A176118 The n-th derivative of 1/x^x, evaluated at x=1.

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A215522 n-th derivative of (x^x)^(x^x) at x=1.

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A007120 Expansion of e.g.f. (1+x)^(1-x).

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

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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.

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A354795 Triangle read by rows. The matrix inverse of A354794. Equivalently, the Bell transform of cfact(n) = -(n - 1)! if n > 0 and otherwise 1/(-n)!.

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