A176118 -id:A176118 - 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)

A008296 Triangle of Lehmer-Comtet numbers of the first kind.

Original entry on oeis.org

1, 1, 1, -1, 3, 1, 2, -1, 6, 1, -6, 0, 5, 10, 1, 24, 4, -15, 25, 15, 1, -120, -28, 49, -35, 70, 21, 1, 720, 188, -196, 49, 0, 154, 28, 1, -5040, -1368, 944, 0, -231, 252, 294, 36, 1, 40320, 11016, -5340, -820, 1365, -987, 1050, 510, 45, 1, -362880, -98208, 34716, 9020, -7645, 3003, -1617, 2970, 825, 55, 1, 3628800

Offset: 1

N. J. A. Sloane

Triangle arising in the expansion of ((1+x)*log(1+x))^n.

Also the Bell transform of (-1)^(n-1)*(n-1)! if n>1 else 1 adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 16 2016

			Triangle begins:
   1;
   1,  1;
  -1,  3,   1;
   2, -1,   6,  1;
  -6,  0,   5, 10,  1;
  24,  4, -15, 25, 15, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139.

Alois P. Heinz, Rows n = 1..141, flattened
H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y = x^x, Rocky Mountain J. Math. Volume 26, Number 2 (1996), 615-625.
Tian-Xiao He and Yuanziyi Zhang, Centralizers of the Riordan Group, arXiv:2105.07262 [math.CO], 2021.
D. H. Lehmer, Numbers Associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, pp. 461-475.

Cf. A039621 (second kind), A354795 (variant), A185164, A005727 (row sums), A298511 (central).
Columns: A045406 (column 2), A347276 (column 3), A345651 (column 4).
Diagonals: A000142, A000217, A059302.
Cf. A176118.

for n from 1 to 20 do for k from 1 to n do
printf(`%d,`, add(binomial(l,k)*k^(l-k)*Stirling1(n,l), l=k..n)) od: od:
# second program:
A008296 := proc(n, k) option remember; if k=1 and n>1 then (-1)^n*(n-2)! elif n=k then 1 else (n-1)*procname(n-2, k-1) + (k-n+1)*procname(n-1, k) + procname(n-1, k-1) end if end proc:
seq(print(seq(A008296(n, k), k=1..n)), n=1..7); # Mélika Tebni, Aug 22 2021

a[1, 1] = a[2, 1] = 1; a[n_, 1] = (-1)^n (n-2)!;
a[n_, n_] = 1; a[n_, k_] := a[n, k] = (n-1) a[n-2, k-1] + a[n-1, k-1] + (k-n+1) a[n-1,k]; Flatten[Table[a[n, k], {n, 1, 12}, {k, 1, n}]][[1 ;; 67]]
(* Jean-François Alcover, Apr 29 2011 *)

{T(n, k) = if( k<1 || k>n, 0, n! * polcoeff(((1 + x) * log(1 + x + x * O(x^n)))^k / k!, n))}; /* Michael Somos, Nov 15 2002 */

# uses[bell_matrix from A264428]
# Adds 1, 0, 0, 0, ... as column 0 at the left side of the triangle.
bell_matrix(lambda n: (-1)^(n-1)*factorial(n-1) if n>1 else 1, 7) # Peter Luschny, Jan 16 2016

E.g.f. for a(n, k): (1/k!)[ (1+x)*log(1+x) ]^k. - Len Smiley
Left edge is (-1)*n!, for n >= 2. Right edge is all 1's.
a(n+1, k) = n*a(n-1, k-1) + a(n, k-1) + (k-n)*a(n, k).
a(n, k) = Sum_{m} binomial(m, k)*k^(m-k)*Stirling1(n, m).
From Peter Bala, Mar 14 2012: (Start)
E.g.f.: exp(t*(1 + x)*log(1 + x)) = Sum_{n>=0} R(n,t)*x^n/n! = 1 + t*x + (t+t^2)x^2/2! + (-t+3*t^2+t^3)x^3/3! + .... Cf. A185164. The row polynomials R(n,t) are of binomial type and satisfy the recurrence R(n+1,t) = (t-n)*R(n,t) + t*d/dt(R(n,t)) + n*t*R(n-1,t) with R(0,t) = 1 and R(1,t) = t. Inverse array is A039621.
(End)
Sum_{k=0..n} (-1)^k * a(n,k) = A176118(n). - Alois P. Heinz, Aug 25 2021

More terms from James Sellers, Jan 26 2001

Edited by N. J. A. Sloane at the suggestion of Andrew Robbins, Dec 11 2007

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

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))))).

A350759 a(n) = Sum_{k=0..n} (-1)^kA345652(k)Stirling1(n, k).

Original entry on oeis.org

1, 0, -1, 1, 1, -4, 1, 29, -167, 1000, -7989, 75857, -794639, 9058180, -111944923, 1492748581, -21369667087, 326932765840, -5323818187817, 91947960224097, -1678914212753599, 32317295442288844, -654084630476955479, 13886774070229667213

Offset: 0

Mélika Tebni, Jan 14 2022

sign

Conjectures: For all p prime, (a(p) + a(p+1) - 2) == 0 (mod p),

a(p+1) == 1 (mod ((p+1)*p)).

			a(9) = -Sum_{k=0..7} a(k)*A238363(8, k).
a(9) = -(1*(-5040) + 0*(5760) - 1*(-3360) + 1*(1344) + 1*(-420) - 4*(112) + 1*(-28) + 29*(8)) = 1000.
E.g.f.: 1 - x^2/2! + x^3/3! + x^4/4! - 4*x^5/5! + x^6/6! + 29*x^7/7! - 167*x^8/8! + 1000*x^9/9! + ...

Cf. A048994, A176118, A238363, A345652.

b := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*b(k), k=0..n-2)) end:
a := n-> add((-1)^k*b(k)*Stirling1(n, k), k=0..n):
seq(a(n), n=0..23);
# Second program:
a := proc(n) option remember; `if`(n=0, 1, add((n-2-k)!*binomial(n-1, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end:
seq(a(n), n=0..23);
# Third program:
a := series(exp(-1+(1+x)*(1-log(1+x))), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# Fourth program:
A350759 := n-> add(binomial(n, k)*(n-k)!*coeftayl(x^(-x), x=1, n-k), k=0..n):
seq(A350759 (n), n=0..23); # Mélika Tebni, Mar 31 2022

CoefficientList[Series[Exp[-1+(1+x)*(1-Log[1+x])], {x, 0, 23}], x] * Range[0, 23]!

my(x='x+O('x^30)); Vec(serlaplace(exp(-1 + (1 + x)*(1 - log(1 + x))))) \\ Michel Marcus, Jan 14 2022

from math import comb, factorial
def a(n):
    return 1 if n == 0 else sum([factorial(n-2-k)*comb(n-1, k)*(-1)**(n-1-k)*a(k) for k in range(n-1)])
print([a(n) for n in range(24)])

a(0) = 1, a(n) = -Sum_{k=0..n-2} a(k)*A238363(n-1, k) for n > 0.
a(0) = 1, a(n) = Sum_{k=0..n-2} (n-2-k)!*binomial(n-1, k)*(-1)^(n-1-k)*a(k) for n > 0.
E.g.f.: exp(-1 + (1 + x)*(1 - log(1 + x))).
E.g.f. y(x) satisfies y' + y*log(1 + x) = 0.
a(n) = Sum_{k=0..n} binomial(n, k)*A176118(n-k). - Mélika Tebni, Mar 31 2022
a(n) ~ -(-1)^n * n! * exp(-1) / n^2 * (1 - 2*log(n)/n). - Vaclav Kotesovec, Mar 31 2022

A366228 Expansion of e.g.f. A(x) satisfying A(x) = 1 + Integral A(x)^A(x) dx.

Original entry on oeis.org

1, 1, 1, 3, 12, 68, 473, 3998, 39327, 443599, 5629807, 79486044, 1235018598, 20946691457, 385025599130, 7624623236381, 161823815625933, 3664505951884255, 88189911547566082, 2247691180645108608, 60480432646998315279, 1713328345952593367876, 50970518521542636421145

Offset: 0

Paul D. Hanna, Nov 13 2023

nonn

(a(n)/(n-1)!)^(1/n) tends to 1.42011... - Vaclav Kotesovec, Nov 15 2023

			E.g.f.: A(x) = 1 + x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 68*x^5/5! + 473*x^6/6! + 3998*x^7/7! + 39327*x^8/8! + 443599*x^9/9! + 5629807*x^10/10! + ...
where A(x) = 1 + Integral A(x)^A(x) dx.
RELATED SERIES.
A(x)^A(x) = 1 + x + 3*x^2/2! + 12*x^3/3! + 68*x^4/4! + 473*x^5/5! + 3998*x^6/6! + 39327*x^7/7! + 443599*x^8/8! + ...
log(A(x)) = x + 2*x^3/3! + 3*x^4/4! + 32*x^5/5! + 155*x^6/6! + 1575*x^7/7! + 13573*x^8/8! + 160756*x^9/9! + 1938288*x^10/10! + ...
A(x)^(A(x) - 1) = 1 + 2*x^2/2! + 3*x^3/3! + 32*x^4/4! + 155*x^5/5! + 1575*x^6/6! + 13573*x^7/7! + ...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A194786, A176118.

{a(n) = my(A=1); for(i=0, n, A = 1 + intformal( A^A +x*O(x^n) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = my(A=1); for(i=0, n, A = exp( intformal( A^(A-1) +x*O(x^n) ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = 1 + Integral A(x)^A(x) dx.
(2) A(x) = exp( Integral A(x)^(A(x) - 1) dx ).
(3) A(x) = 1 + Series_Reversion( Integral 1/(1+x)^(1+x) dx ), where 1/(1+x)^(1+x) is the e.g.f. of A176118.
(4) A(x)^A(x) = 1/Sum_{n>=0} (1 - A(x))^n/n! * Product_{k=1..n} (k + A(x)-1) = A'(x). - Paul D. Hanna, Jul 25 2025

cp's OEIS Frontend

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

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A008296 Triangle of Lehmer-Comtet numbers of the first kind.

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

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A350759 a(n) = Sum_{k=0..n} (-1)^k*A345652(k)*Stirling1(n, k).

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A366228 Expansion of e.g.f. A(x) satisfying A(x) = 1 + Integral A(x)^A(x) dx.

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A350759 a(n) = Sum_{k=0..n} (-1)^kA345652(k)Stirling1(n, k).