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A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.

%I A038037 #49 Sep 17 2022 03:12:35
%S A038037 1,2,9,68,730,10164,173838,3524688,82627200,2198295360,65431163160,
%T A038037 2154106470240,77714083773456,3048821300491680,129221979665461200,
%U A038037 5884296038166954240,286492923374605966080,14851359950834255500800
%N A038037 Number of labeled rooted compound windmills (mobiles) with n nodes.
%D A038037 F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).
%H A038037 Alois P. Heinz, <a href="/A038037/b038037.txt">Table of n, a(n) for n = 1..140</a>
%H A038037 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A038037 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 454
%H A038037 B. R. Jones, <a href="http://summit.sfu.ca/item/14554">On tree hook length formulas, Feynman rules and B-series</a>, Master's thesis, Simon Fraser University, 2014.
%H A038037 <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>
%F A038037 Divides by n and shifts left under "CIJ" (necklace, indistinct, labeled) transform.
%F A038037 E.g.f. A(x) satisfies A(x) = x-x*log(1-A(x)). [Corrected by _Andrey Zabolotskiy_, Sep 16 2022]
%F A038037 a(n) = Sum_{j=0..n} binomial(n,j)*abs(Stirling1(n-1,j))*j!, n > 0. - _Vladimir Kruchinin_, Feb 03 2011
%F A038037 a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^(n-1) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Dec 27 2013
%F A038037 E.g.f.: series reversion of x/(1 - log(1-x)). - _Andrew Howroyd_, Sep 19 2018
%p A038037 logtr:= proc(p) local b; b:=proc(n) option remember; local k; if n=0 then 1 else p(n)- add(k *binomial(n,k) *p(n-k) *b(k), k=1..n-1)/n fi end end: b:= logtr(-a): a:= n-> `if`(n<=1,1, -n*b(n-1)): seq(a(n), n=1..25); # _Alois P. Heinz_, Sep 14 2008
%t A038037 a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
%t A038037 (* _Jean-François Alcover_, Jun 22 2011, after _Vladimir Kruchinin_ *)
%o A038037 (PARI) Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ _Andrew Howroyd_, Sep 19 2018
%Y A038037 Cf. A029768, A032200, A055349.
%K A038037 nonn,eigen
%O A038037 1,2
%A A038037 _Christian G. Bower_, Sep 15 1998