A009444 E.g.f. log(1 + x*exp(-x)).

0, 1, -3, 11, -58, 409, -3606, 38149, -470856, 6641793, -105398650, 1858413061, -36044759796, 762659322385, -17481598316742, 431535346662645, -11413394655983536, 321989729198400385, -9651573930139850610

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

abs(a(n)) is the number of connected functions f:{1,2,...,n}->{1,2,...,n} such that every element is mapped into a recurrent element. Cf. A006153. - Geoffrey Critzer, May 24 2012

Links

• G. C. Greubel, Table of n, a(n) for n = 0..400
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 125

Crossrefs

Cf. A006153, A009306, A305990.

Programs

Mathematica

With[{nmax = 40}, CoefficientList[Series[Log[1 + x*Exp[-x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Nov 22 2017 *)

Maxima

a(n):=(-1)^(n+1)*n!*sum(m^(n-m-1)/(n-m)!,m,1,n); /* Vladimir Kruchinin, Oct 08 2011 */

PARI

x='x+O('x^66); /* that many terms */
egf=1/(1+x/exp(x)); /* = 1 - x + 2*x^2 - 7/2*x^3 + 37/6*x^4 - 87/8*x^5 +... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, Apr 30 2011 */

Sage

A009444 = lambda n: (-1)^(n+1)*factorial(n)*sum(m^(n-m-1)/factorial(n-m) for m in (1..n))
[A009444(n) for n in (0..9)] # Peter Luschny, Jan 18 2016

Formula

abs(a(n)) is asymptotic to (n-1)!/LambertW(1)^n. - Vladeta Jovovic, Jul 12 2007

Sequence of absolute values has e.g.f. log(1/(1-x*exp(x))). - Joerg Arndt, Apr 30 2011

a(n) = (-1)^(n+1)*n!*sum(m=1..n, m^(n-m-1)/(n-m)!). - Vladimir Kruchinin, Oct 08 2011

a(n) = (-1)^(n + 1) * n + Sum_{k=1..n-1} (-1)^(n - k) * binomial(n-1,k-1) * (n - k) * a(k). - Ilya Gutkovskiy, Jan 17 2020

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, Apr 30 2011