cp's OEIS Frontend

A123916 Number of binary words whose (unique) decreasing Lyndon decomposition is into Lyndon words each with an odd number of 1's; EULER transform of A000048.

%I A123916 #21 Aug 16 2022 16:06:29
%S A123916 1,1,2,3,6,10,19,34,65,120,229,432,829,1583,3051,5874,11370,22012,
%T A123916 42756,83113,161917,315723,616588,1205232,2358604,4619485,9055960,
%U A123916 17766086,34880215,68524486,134707150,264960828,521449025
%N A123916 Number of binary words whose (unique) decreasing Lyndon decomposition is into Lyndon words each with an odd number of 1's; EULER transform of A000048.
%H A123916 Vaclav Kotesovec, <a href="/A123916/b123916.txt">Table of n, a(n) for n = 1..700</a>
%F A123916 Prod_{n>=1} 1/(1-q^n)^A000048(n) = 1 + sum_{n>=1} a(n) q^n.
%F A123916 G.f. A(x) satisfies: A(x)^2 = A(x^2) / (1 - 2*x). - _Paul D. Hanna_, Apr 17 2016
%F A123916 a(n) ~ c * 2^n / sqrt(n), where c = 0.3412831644583761326654... . - _Vaclav Kotesovec_, Apr 18 2016
%F A123916 G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (v^2 - 2*u^2*v - u^4) + 2*w*u^4. - _Michael Somos_, Jun 27 2017
%e A123916 The binary words 1111, 1101, 1001, 0101, 0111, 0001 of length 4 decompose as 1*1*1*1, 1*1*01, 1*001, 01*01, 0111, 0001 and each subword has an odd number of 1's, therefore a(4)=6.
%e A123916 G.f. A(x) = x + x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 10*x^6 + 19*x^7 + 34*x^8 + ... such that A(x)^2 * (1 - 2*x) = A(x^2).
%o A123916 (PARI) /* G.f. A(x) satisfies: A(x)^2 = A(x^2)/(1 - 2*x) */
%o A123916 {a(n) = my(A=x); for(i=1,n, A = sqrt( subst(A,x, x^2)/(1 - 2*x +x*O(x^n)))); polcoeff(A,n)}
%o A123916 for(n=1,50, print1(a(n),", ")) \\ _Paul D. Hanna_, Apr 17 2016
%o A123916 (PARI) /* As the EULER transform of A000048 */
%o A123916 {A000048(n) = sumdiv(n, d, (d%2)*(moebius(d)*2^(n/d)))/(2*n)} \\ _Michael B. Porter_
%o A123916 {a(n) = polcoeff( prod(k=1,n, 1/(1 - x^k +x*O(x^n))^A000048(k)), n-1)}
%o A123916 for(n=1,50, print1(a(n),", ")) \\ _Paul D. Hanna_, Apr 17 2016
%Y A123916 Cf. A000048, A271929.
%K A123916 nonn
%O A123916 1,3
%A A123916 _Mike Zabrocki_, Oct 28 2006