cp's OEIS Frontend

A123915 Number of binary words whose (unique) decreasing Lyndon decomposition is into Lyndon words each with an even number of 1's; EULER transform of A051841.

%I A123915 #10 Nov 19 2020 12:00:27
%S A123915 1,1,1,2,3,6,11,21,39,75,143,275,528,1020,1971,3821,7414,14419,28072,
%T A123915 54739,106847,208815,408470,799806,1567333,3073916,6032971,11848693,
%U A123915 23285202,45787650,90085410,177331748,349243800,688129474,1356433342,2674877358,5276869233
%N A123915 Number of binary words whose (unique) decreasing Lyndon decomposition is into Lyndon words each with an even number of 1's; EULER transform of A051841.
%H A123915 Alois P. Heinz, <a href="/A123915/b123915.txt">Table of n, a(n) for n = 0..1000</a>
%F A123915 Prod_{n>=1} 1/(1-q^n)^A051841(n) = 1+sum_{n>=1} a(n) q^n.
%F A123915 a(n) ~ c * 2^n / sqrt(n), where c = 0.466342789995157602308480670781344540837057109916338560252870092619488755668... - _Vaclav Kotesovec_, May 31 2019
%e A123915 The binary words 00000, 01100, 00110, 01111, 00011, 00101 of length 5 decompose as 0*0*0*0*0, 011*0*0, 0011*0, 01111, 00011, 00101 and each subword has an even number of 1's, therefore a(5)=6.
%p A123915 with(numtheory):
%p A123915 b:= proc(n) option remember; add(igcd(d, 2)*
%p A123915       2^(n/d)*mobius(d), d=divisors(n))/(2*n)
%p A123915     end:
%p A123915 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A123915       d*b(d), d=divisors(j))*a(n-j), j=1..n)/n)
%p A123915     end:
%p A123915 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 28 2017
%t A123915 b[n_] := b[n] = Sum[GCD[d, 2] 2^(n/d) MoebiusMu[d], {d, Divisors[n]}]/(2n);
%t A123915 a[n_] := a[n] = If[n==0, 1, Sum[Sum[d b[d], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
%t A123915 a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 19 2020, after _Alois P. Heinz_ *)
%Y A123915 Cf. A051841.
%K A123915 nonn
%O A123915 0,4
%A A123915 _Mike Zabrocki_, Oct 28 2006
%E A123915 a(0)=1 prepended by _Alois P. Heinz_, Jul 28 2017