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

Author

Christian G. Bower

Keywords

• nonn
• eigen

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

Examples

			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)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Wikipedia, Lyndon word
• Index entries for sequences related to mobiles

Crossrefs

Cf. A032200, A055363.

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

Programs

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 *)

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

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)