%I #35 Nov 19 2023 10:34:10

%S 1,-1,0,3,-8,10,6,-42,-160,2952,-27720,253440,-2553528,28562664,

%T -349272000,4618376280,-65615072640,996952226880,-16133983959744,

%U 277093189849536,-5033937521116800,96451913892983040,-1943937259314019200,41112770486238380160

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

%H Alois P. Heinz, <a href="/A176118/b176118.txt">Table of n, a(n) for n = 0..450</a>

%F 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

%F a(n) ~ (-1)^(n+1) * n! / n^2. - _Vaclav Kotesovec_, Sep 03 2014

%F E.g.f.: 1/(x+1)^(x+1). - _Alois P. Heinz_, Sep 27 2016

%F a(n) = Sum_{k=0..n} (-1)^k * A008296(n,k). - _Alois P. Heinz_, Aug 25 2021

%F E.g.f.: Sum_{n>=0} (-1)^n * x^n/n! * Product_{k=1..n} (k + x). - _Paul D. Hanna_, Nov 13 2023

%e 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! + ...

%e The e.g.f. as a power series with reduced fractional coefficients begins

%e 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 + ...

%p 1, seq(simplify(subs(x = 1, diff(x^(-x), `$`(x, n)))), n = 1 .. 22); # _Emeric Deutsch_, Apr 14 2010

%p a:= n-> n! *coeftayl(x^(-x), x=1, n):

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 18 2012

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

%Y Cf. A005727, A008296, A194786.

%K sign

%O 0,4

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

%E Definition edited by _Emeric Deutsch_, Apr 14 2010

%E More terms from _Emeric Deutsch_ and _R. J. Mathar_, Apr 14 2010