A069856 - OEIS

%I #28 Dec 20 2021 04:13:12

%S 1,0,3,-17,169,-2079,31261,-554483,11336753,-262517615,6791005621,

%T -194103134499,6074821125385,-206616861429575,7588549099814957,

%U -299320105069298459,12619329503201165281,-566312032570838608863,26952678355224681891685

%N E.g.f.: exp(x)/(1+LambertW(x)).

%C Inverse binomial transform of A000312. - _Tilman Neumann_, Dec 13 2008

%C The |a(n)| is the number of functions f:{1,2,...,n}->{1,2,...,n} such that the digraph representation of f has no isolated vertices. (* _Geoffrey Critzer_, Nov 13 2011 *)

%D sci.math article 3CBC2B66.224E(AT)olympus.mons

%H Vincenzo Librandi, <a href="/A069856/b069856.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = n! * Sum_{k=0..n} (-1)^k*k^k/(k!*(n - k)!).

%F E.g.f. for absolute value of {a(n)}: exp(C(x)-x) where C(x) is the e.g.f for A001865. - _Geoffrey Critzer_, Nov 13 2011, corrected by _Vaclav Kotesovec_, Nov 27 2012

%F abs(a(n)) ~ (exp(1)*n-1/2)/exp(1+exp(-1)) * n^(n-1). - _Vaclav Kotesovec_, Nov 27 2012

%F a(n) = (-1)^n * A350212(n,0). - _Alois P. Heinz_, Dec 19 2021

%t t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Range[0, 20]! CoefficientList[Series[Exp[-x]/(1 - t), {x, 0, 20}], x] (* _Geoffrey Critzer_, Nov 13 2011 *)

%t Range[0, 18]! CoefficientList[ Series[ Exp[x]/(1 + LambertW[x]), {x, 0, 18}], x] (* _Robert G. Wilson v_, Nov 28 2012 *)

%o (PARI) x='x+O('x^50); Vec(serlaplace(exp(x)/(1+lambertw(x)))) \\ _G. C. Greubel_, Jun 11 2017

%Y Cf. A086331.

%Y Cf. A350212.

%K sign

%O 0,3

%A Joe Keane (jgk(AT)jgk.org), May 03 2002