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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. 2
 1, 2, 8, 53, 497, 6027, 89595, 1576682, 32047986, 738772383, 19042778713, 542704904381, 16944005908637, 575128775147734, 21086473359281088, 830481043455973053, 34967280863073327597, 1567405219938012472847, 74521905471659239870631, 3745801599865304794344662 (list; graph; refs; listen; history; text; internal format)
 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 Sequence in context: A323871 A183945 A193651 * A203109 A197795 A327354 Adjacent sequences: A195976 A195977 A195978 * A195980 A195981 A195982 KEYWORD nonn AUTHOR N. J. A. Sloane, Sep 25 2011 STATUS approved

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Last modified August 8 15:25 EDT 2024. Contains 375022 sequences. (Running on oeis4.)