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

0,4

Links

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

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

Example

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 + ...

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 *)

Crossrefs

Cf. A005727, A008296, A194786.

Sequence in context: A095866 A067569 A127518 * A151693 A007284 A080892

Adjacent sequences: A176115 A176116 A176117 * A176119 A176120 A176121

Keyword

sign

Author

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

Extensions

Definition edited by Emeric Deutsch, Apr 14 2010

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

Status

approved