A195979 a(n) = Sum_{k=0..n} T(n,k), where T(n,k) is the number of rooted labeled trees with n nodes whose maximal decreasing subtree has k nodes.

1, 2, 8, 53, 497, 6027, 89595, 1576682, 32047986, 738772383, 19042778713, 542704904381, 16944005908637, 575128775147734, 21086473359281088, 830481043455973053, 34967280863073327597, 1567405219938012472847, 74521905471659239870631, 3745801599865304794344662

Offset: 0

Author

N. J. A. Sloane, Sep 25 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386
• S. Seo and H. Shin, Another refinement for Rooted Trees, arXiv preprint arXiv:1106.1290 [math.CO], 2011-2012.

Programs

Maple

T:= (n, k)-> add(binomial(n+1, m+1) *Stirling2(m+1, k+1)
             *(n-k)^(n-m-1) *(m-k), m=k..n):
a:= n-> 1 +add(T(n, k), k=0..n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 30 2011
# second Maple program:
b:= proc(n, k) option remember;
     `if`(n=0, 1, k*b(n-1, k)+b(n-1, k-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..26);  # Alois P. Heinz, Aug 04 2021
# e.g.f. Maple program:
A195979 := series(exp(1-exp(LambertW(-x)))/(1+LambertW(-x)), x = 0, 20):
seq(n!*coeff(A195979, x, n), n = 0 .. 19); # Mélika Tebni, Jun 13 2023

Mathematica

T[n_, k_] := Sum[Binomial[n+1, m+1]*StirlingS2[m+1, k+1]*(n-k)^(n-m-1)*(m-k), {m, k, n}]; a[n_] := 1 + Sum[T[n, k], {k, 0, n-1}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 07 2014, after Alois P. Heinz *)

Formula

Seo and Shin give an e.g.f.

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n - k)^n/k!. - Ilya Gutkovskiy, Jun 13 2019

a(n) ~ exp(1-exp(-1)) * n^n. - Vaclav Kotesovec, Aug 04 2021

E.g.f.: exp(1-exp(LambertW(-x))) / (1+LambertW(-x)). - Mélika Tebni, Jun 13 2023