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

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

Author

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

Keywords

• sign

Examples

			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 + ...
		

Links

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

Crossrefs

Cf. A005727, A008296, A194786.

Programs

Maple

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

Mathematica

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

Formula

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

Extensions

Definition edited by Emeric Deutsch, Apr 14 2010

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