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
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