OFFSET
0,3
COMMENTS
Essentially the same as A038049.
Also the number of rooted trees whose nodes are labeled with the blocks of a set partition of {1,2,...,n} each having a distinguished element. (See A000248.)
The bijection is straightforward. The trees correspond to functional digraphs mapping the distinguished elements towards the root. All the elements within each block are mapped to the distinguished element of that block. The distinguished element in the root node is the fixed point.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
FORMULA
E.g.f.: T(x*exp(x)) where T(x) is the e.g.f. for A000169.
a(n) = Sum_{k=1..n} binomial(n,k)*k^(n-1).
a(n) ~ sqrt(1+LambertW(exp(-1))) * n^(n-1) / (exp(n)*LambertW(exp(-1))^n). - Vaclav Kotesovec, Jul 09 2013
O.g.f.: Sum_{n>=0} n^(n-1)* x^n / (1 - n*x)^(n+1). - Paul D. Hanna, May 22 2018
E.g.f.: the compositional inverse of A(x) is -A(-x). - Alexander Burstein, Aug 11 2018
MATHEMATICA
With[{nmax = 20}, CoefficientList[Series[-LambertW[-x*Exp[x]], {x, 0, nmax}], x]*Range[0, nmax]!] (* modified by G. C. Greubel, Nov 15 2017 *)
PROG
(PARI) for(n=0, 30, print1(sum(k=1, n, binomial(n, k)*k^(n-1)), ", ")) \\ G. C. Greubel, Nov 15 2017
(PARI) my(x='x+O('x^50)); concat([0], Vec(serlaplace(-lambertw(-x*exp(x))))) \\ G. C. Greubel, Nov 15 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Sep 17 2012
STATUS
approved