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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.

%I A195979 #47 Aug 03 2025 09:21:05
%S A195979 1,2,8,53,497,6027,89595,1576682,32047986,738772383,19042778713,
%T A195979 542704904381,16944005908637,575128775147734,21086473359281088,
%U A195979 830481043455973053,34967280863073327597,1567405219938012472847,74521905471659239870631,3745801599865304794344662
%N 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.
%H A195979 Alois P. Heinz, <a href="/A195979/b195979.txt">Table of n, a(n) for n = 0..386</a>
%H A195979 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 A195979 Seo and Shin give an e.g.f.
%F A195979 a(n) = exp(1) * Sum_{k>=0} (-1)^k*(n - k)^n/k!. - _Ilya Gutkovskiy_, Jun 13 2019
%F A195979 a(n) ~ exp(1-exp(-1)) * n^n. - _Vaclav Kotesovec_, Aug 04 2021
%F A195979 E.g.f.: exp(1-exp(LambertW(-x))) / (1+LambertW(-x)). - _Mélika Tebni_, Jun 13 2023
%p A195979 T:= (n, k)-> add(binomial(n+1, m+1) *Stirling2(m+1, k+1)
%p A195979              *(n-k)^(n-m-1) *(m-k), m=k..n):
%p A195979 a:= n-> 1 +add(T(n, k), k=0..n-1):
%p A195979 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 30 2011
%p A195979 # second Maple program:
%p A195979 b:= proc(n, k) option remember;
%p A195979      `if`(n=0, 1, k*b(n-1, k)+b(n-1, k-1))
%p A195979     end:
%p A195979 a:= n-> b(n$2):
%p A195979 seq(a(n), n=0..26);  # _Alois P. Heinz_, Aug 04 2021
%p A195979 # e.g.f. Maple program:
%p A195979 A195979 := series(exp(1-exp(LambertW(-x)))/(1+LambertW(-x)), x = 0, 20):
%p A195979 seq(n!*coeff(A195979, x, n), n = 0 .. 19); # _Mélika Tebni_, Jun 13 2023
%t A195979 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 A195979 nonn
%O A195979 0,2
%A A195979 _N. J. A. Sloane_, Sep 25 2011