A069856 E.g.f.: exp(x)/(1+LambertW(x)).

1, 0, 3, -17, 169, -2079, 31261, -554483, 11336753, -262517615, 6791005621, -194103134499, 6074821125385, -206616861429575, 7588549099814957, -299320105069298459, 12619329503201165281, -566312032570838608863, 26952678355224681891685

Offset: 0

Author

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

Keywords

• sign

Comments

Inverse binomial transform of A000312. - Tilman Neumann, Dec 13 2008

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

References

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A086331.

Cf. A350212.

Programs

Mathematica

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

PARI

my(x='x+O('x^20)); Vec(serlaplace(exp(x)/(1+lambertw(x)))) \\ G. C. Greubel, Jun 11 2017

Formula

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

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

abs(a(n)) ~ (exp(1)*n-1/2)/exp(1+exp(-1)) * n^(n-1). - Vaclav Kotesovec, Nov 27 2012

a(n) = (-1)^n * A350212(n,0). - Alois P. Heinz, Dec 19 2021