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A052506 Expansion of e.g.f. exp(x*exp(x)-x). 20
1, 0, 2, 3, 16, 65, 336, 1897, 11824, 80145, 586000, 4588001, 38239224, 337611001, 3144297352, 30779387745, 315689119456, 3383159052833, 37790736663456, 439036039824193, 5294386116882280, 66155074120062921, 855156188538926296, 11418964004032623809 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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

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 *)

PROG

(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

(Magma) m:=30; R<x>:=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

(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

CROSSREFS

Cf. A000248, A240989.

Cf. A351736, A351737.

Sequence in context: A012700 A012705 A103331 * A355229 A052858 A191416

Adjacent sequences:  A052503 A052504 A052505 * A052507 A052508 A052509

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified October 6 12:35 EDT 2022. Contains 357264 sequences. (Running on oeis4.)