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

1, 2, 9, 68, 730, 10164, 173838, 3524688, 82627200, 2198295360, 65431163160, 2154106470240, 77714083773456, 3048821300491680, 129221979665461200, 5884296038166954240, 286492923374605966080, 14851359950834255500800

Offset: 1

Author

Christian G. Bower, Sep 15 1998

Keywords

• nonn
• eigen

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.83).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..140
• C. G. Bower, Transforms (2)
• P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 454
• B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
• Index entries for sequences related to mobiles

Crossrefs

Cf. A029768, A032200, A055349.

Programs

Maple

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

Mathematica

a[n_] = Sum[Binomial[n, j]*Abs[StirlingS1[n-1, j]]*j!, {j, 0, n}]; Array[a, 18]
(* Jean-François Alcover, Jun 22 2011, after Vladimir Kruchinin *)

PARI

Vec(serlaplace(serreverse(x/(1 - log(1-x + O(x^20)))))) \\ Andrew Howroyd, Sep 19 2018

Formula

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