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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1667

Index entries for sequences related to Lyndon words

Formula

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

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

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 *)

Crossrefs

Cf. A022553, A289978.

Sequence in context: A149354 A197659 A360468 * A003444 A220881 A149355

Adjacent sequences: A292284 A292285 A292286 * A292288 A292289 A292290

Keyword

nonn

Author

Alois P. Heinz, Sep 20 2017

Status

approved