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

%I A292287 #38 Nov 23 2023 10:36:59
%S A292287 1,1,4,12,43,142,508,1781,6414,23124,84296,308613,1137129,4207456,
%T A292287 15636927,58322808,218272766,819319778,3083913810,11636761924,
%U A292287 44010780075,166802192488,633420816341,2409731688860,9182826866499,35048239457878,133965833871427
%N A292287 Number of multisets of exactly n nonempty balanced binary Lyndon words with a total of 4n letters (2n zeros and 2n ones).
%H A292287 Alois P. Heinz, <a href="/A292287/b292287.txt">Table of n, a(n) for n = 0..1667</a>
%H A292287 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A292287 G.f.: Product_{j>=1} 1/(1-x^j)^A022553(j+1).
%F A292287 a(n) = A289978(2n,n).
%p A292287 with(numtheory):
%p A292287 g:= proc(n) option remember; `if`(n=0, 1, add(
%p A292287       mobius(n/d)*binomial(2*d, d), d=divisors(n))/(2*n))
%p A292287     end:
%p A292287 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A292287       d*g(d+1), d=divisors(j))*a(n-j), j=1..n)/n)
%p A292287     end:
%p A292287 seq(a(n), n=0..30);
%t A292287 g[n_] := g[n] = If[n == 0, 1, Sum[MoebiusMu[n/d] Binomial[2d, d], {d, Divisors[n]}]/(2n)];
%t A292287 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 A292287 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 23 2023, after _Alois P. Heinz_ *)
%Y A292287 Cf. A022553, A289978.
%K A292287 nonn
%O A292287 0,3
%A A292287 _Alois P. Heinz_, Sep 20 2017