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A038037
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Number of labeled rooted compound windmills (mobiles) with n nodes.
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11
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1, 2, 9, 68, 730, 10164, 173838, 3524688, 82627200, 2198295360, 65431163160, 2154106470240, 77714083773456, 3048821300491680, 129221979665461200, 5884296038166954240, 286492923374605966080, 14851359950834255500800
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OFFSET
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1,2
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REFERENCES
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F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).
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LINKS
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FORMULA
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Divides by n and shifts left under "CIJ" (necklace, indistinct, labeled) transform.
E.g.f. A(x) satisfies A(x) = x-x*log(1-A(x)). [Corrected by Andrey Zabolotskiy, Sep 16 2022]
a(n) = Sum_{j=0..n} binomial(n,j)*abs(Stirling1(n-1,j))*j!, n > 0. - Vladimir Kruchinin, Feb 03 2011
a(n) ~ sqrt(-1-LambertW(-1,-exp(-2))) * (-LambertW(-1,-exp(-2)))^(n-1) * n^(n-1) / exp(n). - Vaclav Kotesovec, Dec 27 2013
E.g.f.: series reversion of x/(1 - log(1-x)). - Andrew Howroyd, Sep 19 2018
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MAPLE
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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
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MATHEMATICA
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a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
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PROG
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(PARI) Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ Andrew Howroyd, Sep 19 2018
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CROSSREFS
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KEYWORD
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nonn,eigen
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AUTHOR
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STATUS
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approved
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