A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

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

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-François 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 A354076 A151256

Adjacent sequences: A032168 A032169 A032170 * A032172 A032173 A032174

Keyword

nonn,eigen

Author

Christian G. Bower

Status

approved