A052506 Expansion of e.g.f. exp(x*exp(x)-x).

1, 0, 2, 3, 16, 65, 336, 1897, 11824, 80145, 586000, 4588001, 38239224, 337611001, 3144297352, 30779387745, 315689119456, 3383159052833, 37790736663456, 439036039824193, 5294386116882280, 66155074120062921, 855156188538926296, 11418964004032623809

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

a(n) is the number of forests of rooted labeled trees with height exactly one. Equivalently, the number of idempotent mappings from {1,2,...,n} into {1,2,...,n} where each fixed point has at least one (other than itself) element mapped to it. See the second summation formula provided by Vladeta Jovovic with conditions on k, the number of fixed points. - Geoffrey Critzer, Sep 20 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 39
• Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Crossrefs

Cf. A000248, A240989.

Cf. A351736, A351737.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x*Exp(x)-x) )); [Factorial(n-1)*b[n]: n in [1..m-1]]; // G. C. Greubel, May 13 2019

Maple

spec := [S,{S=Set(Tree), Tree=Prod(Z,Set(Z,0 < card))},labeled]: seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

nn=20;Range[0,nn]! CoefficientList[Series[Exp[x(Exp[x]-1)], {x,0,nn}], x]  (* Geoffrey Critzer, Sep 20 2012 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace( exp(x*exp(x)-x) )) \\ G. C. Greubel, Nov 15 2017

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k-1)*x)^(k+1))) \\ Seiichi Manyama, Aug 29 2022

Sage

m = 30; T = taylor(exp(x*exp(x)-x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*(n-k-1)^k. - Vladeta Jovovic, Apr 12 2003

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*k!*Stirling2(n-k, k). - Vladeta Jovovic, Dec 19 2004

a(n) ~ exp((1-r*(n+r))/(1+r)) * n^(n+1/2) * sqrt(1+r) / (r^n * sqrt((1+r)^3 + n*(1+3*r+r^2))), where r satisfies exp(r)*(1+r) - (1+n)/r = 1. - Vaclav Kotesovec, Aug 04 2014

(a(n)/n!)^(1/n) ~ exp(1/(2*LambertW(sqrt(n)/2))) / (2*LambertW(sqrt(n)/2)). - Vaclav Kotesovec, Aug 06 2014

G.f.: Sum_{k>=0} x^k / (1 - (k-1)*x)^(k+1). - Seiichi Manyama, Aug 29 2022