A292287 - OEIS

%I #38 Nov 23 2023 10:36:59

%S 1,1,4,12,43,142,508,1781,6414,23124,84296,308613,1137129,4207456,

%T 15636927,58322808,218272766,819319778,3083913810,11636761924,

%U 44010780075,166802192488,633420816341,2409731688860,9182826866499,35048239457878,133965833871427

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

%H Alois P. Heinz, <a href="/A292287/b292287.txt">Table of n, a(n) for n = 0..1667</a>

%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

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

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

%p with(numtheory):

%p g:= proc(n) option remember; `if`(n=0, 1, add(

%p       mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n))

%p     end:

%p a:= proc(n) option remember; `if`(n=0, 1, add(add(

%p       d*g(d+1), d=divisors(j))*a(n-j), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..30);

%t g[n_] := g[n] = If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/(2n)];

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*g[d+1], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 23 2023, after _Alois P. Heinz_ *)

%Y Cf. A022553, A289978.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 20 2017