A292287 Number of multisets of exactly n nonempty balanced binary Lyndon words with a total of 4n letters (2n zeros and 2n ones).

1, 1, 4, 12, 43, 142, 508, 1781, 6414, 23124, 84296, 308613, 1137129, 4207456, 15636927, 58322808, 218272766, 819319778, 3083913810, 11636761924, 44010780075, 166802192488, 633420816341, 2409731688860, 9182826866499, 35048239457878, 133965833871427

Offset: 0

Author

Alois P. Heinz, Sep 20 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1667
• Index entries for sequences related to Lyndon words

Crossrefs

Cf. A022553, A289978.

Programs

Maple

with(numtheory):
g:= proc(n) option remember; `if`(n=0, 1, add(
      mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*g(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..30);

Mathematica

g[n_] := g[n] = If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/(2n)];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2023, after Alois P. Heinz *)

Formula

G.f.: Product_{j>=1} 1/(1-x^j)^A022553(j+1).

a(n) = A289978(2n,n).