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%I #35 Jun 16 2023 01:59:04
%S 1,2,8,53,497,6027,89595,1576682,32047986,738772383,19042778713,
%T 542704904381,16944005908637,575128775147734,21086473359281088,
%U 830481043455973053,34967280863073327597,1567405219938012472847,74521905471659239870631,3745801599865304794344662
%N 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.
%H Alois P. Heinz, <a href="/A195979/b195979.txt">Table of n, a(n) for n = 0..386</a>
%H S. Seo and H. Shin, <a href="http://arxiv.org/abs/1106.1290">Another refinement for Rooted Trees</a>, arXiv preprint arXiv:1106.1290 [math.CO], 2011-2012.
%F Seo and Shin give an e.g.f.
%F a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n - k)^n/k!. - _Ilya Gutkovskiy_, Jun 13 2019
%F a(n) ~ exp(1-exp(-1)) * n^n. - _Vaclav Kotesovec_, Aug 04 2021
%F E.g.f.: exp(1-exp(LambertW(-x))) / (1+LambertW(-x)). - _Mélika Tebni_, Jun 13 2023
%p T:= (n, k)-> add(binomial(n+1, m+1) *Stirling2(m+1, k+1)
%p *(n-k)^(n-m-1) *(m-k), m=k..n):
%p a:= n-> 1 +add(T(n, k), k=0..n-1):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 30 2011
%p # second Maple program:
%p b:= proc(n, k) option remember;
%p `if`(n=0, 1, -k*b(n-1, k)+b(n-1, k+1))
%p end:
%p a:= n-> b(n, -n):
%p seq(a(n), n=0..26); # _Alois P. Heinz_, Aug 04 2021
%p # e.g.f. Maple program:
%p A195979 := series(exp(1-exp(LambertW(-x)))/(1+LambertW(-x)), x = 0, 20):
%p seq(n!*coeff(A195979, x, n), n = 0 .. 19); # _Mélika Tebni_, Jun 13 2023
%t 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_ *)
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Sep 25 2011
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