A293297 -id:A293297 - 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)

A293472 Triangle read by rows, coefficients of polynomials in t = log(x) of the n-th derivative of x^x, evaluated at x = 1. T(n, k) with n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 8, 12, 12, 4, 1, 10, 40, 30, 20, 5, 1, 54, 60, 120, 60, 30, 6, 1, -42, 378, 210, 280, 105, 42, 7, 1, 944, -336, 1512, 560, 560, 168, 56, 8, 1, -5112, 8496, -1512, 4536, 1260, 1008, 252, 72, 9, 1

Offset: 0

Peter Luschny, Oct 10 2017

			Triangle starts:
0: [  1]
1: [  1,   1]
2: [  2,   2,   1]
3: [  3,   6,   3,   1]
4: [  8,  12,  12,   4,   1]
5: [ 10,  40,  30,  20,   5,  1]
6: [ 54,  60, 120,  60,  30,  6, 1]
7: [-42, 378, 210, 280, 105, 42, 7, 1]
...
For n = 3, the 3rd derivative of x^x is p(3,x,t) = x^x*t^3 + 3*x^x*t^2 + 3*x^x*t + x^x + 3*x^x*t/x + 3*x^x/x - x^x/x^2 where log(x) is substituted by t. Evaluated at x = 1: p(3,1,t) = 3 + 6*t + 3*t^2 + t^3 with coefficients [3, 6, 3, 1].

More generally, consider the n-th derivative of x^(x^m). This is case m = 1.
m      |  t = -1 |  t = 0  |  t = 1  | p(n, t) | related
m = 1  | A203852 | A005727 | A293297 | A293472 | A293239
m = 2  |    -    | A215524 |    -    | A293473 | A290268
m = 3  |    -    | A215704 |    -    | A293474 |    -
Cf. A215703.

dx := proc(m, n) if n = 0 then return [1] fi;
subs(ln(x) = t, diff(x^(x^m), x$n)): subs(x = 1, %):
PolynomialTools:-CoefficientList(%,t) end:
ListTools:-Flatten([seq(dx(1, n), n=0..10)]);

dx[m_, n_] := ReplaceAll[CoefficientList[ReplaceAll[Expand[D[x^x^m, {x, n}]], Log[x] -> t], t], x -> 1];
Table[dx[1, n], {n, 0, 7}] // Flatten

cp's OEIS Frontend

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

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