OFFSET
0,2
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.
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
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, -n):
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 25 2011
STATUS
approved