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A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries. 7
1, 1, 1, 2, 4, 10, 23, 59, 148, 385, 1006, 2678, 7170, 19421, 52933, 145364, 401421, 1114713, 3109710, 8713076, 24506121, 69168705, 195849114, 556165311, 1583601840, 4520226558, 12931917204, 37075154703 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also the number of locally Lyndon plane trees with n nodes, where a plane tree is locally Lyndon if the sequence of branches directly under any given node is a Lyndon word. - Gus Wiseman, Sep 05 2018

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

Wikipedia, Lyndon word

Index entries for sequences related to mobiles

FORMULA

Shifts left under "CHK" (necklace, identity, unlabeled) transform.

From Petros Hadjicostas, Dec 03 2017: (Start)

a(n+1) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = c(1) = 1.

G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+1)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).

The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 147*x^6 + 414*x^7 + 1203*x^8 + ...

(End)

EXAMPLE

From Gus Wiseman, Sep 05 2018: (Start)

The a(6) = 10 locally Lyndon plane trees:

  (((((o)))))

  (((o(o))))

  ((o((o))))

  (o(((o))))

  ((o)((o)))

  ((oo(o)))

  (o(o(o)))

  (oo((o)))

  (o(o)(o))

  (ooo(o))

(End)

MATHEMATICA

T[n_, k_] := Module[{A}, A[_, _] = 0; If[k < 1 || k > n, 0, For[j = 1, j <= n, j++, A[x_, y_] = x*y - x*Sum[MoebiusMu[i]/i * Log[1 -  A [x^i, y^i]] + O[x]^j // Normal , {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];

a[n_] := a[n] = Sum[T[n, k], {k, 1, n}];

Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 28}] (* Jean-Fran├žois Alcover, Jun 30 2017, using Michael Somos' code for A055363 *)

LyndonQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And]&&Array[RotateRight[q, #]&, Length[q], 1, UnsameQ];

lynplane[n_]:=If[n==1, {{}}, Join@@Table[Select[Tuples[lynplane/@c], LyndonQ], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];

Table[Length[lynplane[n]], {n, 10}] (* Gus Wiseman, Sep 05 2018 *)

PROG

(PARI)

CHK(p, n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}

seq(n)={my(p=O(1)); for(i=1, n, p=1+CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

CROSSREFS

Cf. A032200, A055363.

Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

Sequence in context: A137681 A127389 A152173 * A127713 A151256 A205999

Adjacent sequences:  A032168 A032169 A032170 * A032172 A032173 A032174

KEYWORD

nonn,eigen

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 20 00:56 EDT 2018. Contains 316378 sequences. (Running on oeis4.)