A108529 - OEIS

%I #30 Aug 31 2018 15:57:15

%S 1,1,2,5,16,51,177,621,2246,8245,30783,116257,443945,1710255,6640939,

%T 25961690,102105115,403701135,1603721999,6397931901,25621989760,

%U 102965680728,415091909292,1678226164646,6803121058354,27645628327636

%N Number of asymmetric mobiles (cycle rooted trees) with n generators.

%C A generator is a leaf or a node with just one child.

%C Here CHK(A(x)) = 1 - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)), i.e., the constant 1 is included in the definition of the CHK transform. For other sequences that involve the CHK transform, the 1 is sometimes dropped; e.g., see sequence A032171. We have CHK(A(x)) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 86*x^6 + 303*x^7 + 1065*x^8 + 3871*x^9 + ... - _Petros Hadjicostas_, Dec 05 2017

%H Andrew Howroyd, <a href="/A108529/b108529.txt">Table of n, a(n) for n = 1..200</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>

%H <a href="/index/Mo#mobiles">Index entries for sequences related to mobiles</a>

%F G.f. satisfies: (2-x)*A(x) = x - 1 + CHK(A(x)).

%F From _Petros Hadjicostas_, Dec 05 2017: (Start)

%F a(n) = (1/2)*(a(n-1) + (1/n)*Sum_{d|n} mu(d)*c(n/d)) for n>=2, where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) and a(1) = c(1) = 1.

%F The g.f. satisfies (2-x)*A(x) = x - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)). (This is just a rephrasing of C. Bower's equation above.)

%F The auxiliary sequence (c(n): n>=1} has g.f. C(x) = Sum_{n>=1} c(n)*x^n = x*(dA/dx)/(1-A(x)) = x + 3*x^2 + 10*x^3 + 35*x^4 + 136*x^5 + 528*x^6 + 2122*x^7 + ...

%F (End)

%o (PARI)

%o CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}

%o seq(n)={my(p=x); for(n=2, n, p += x^n*polcoef(x*p + CHK(p, n), n)); Vecrev(p/x)} \\ _Andrew Howroyd_, Aug 31 2018

%Y Cf. A108521, A108522, A108523, A108524, A108525, A108527, A108528, A032171.

%K nonn

%O 1,3

%A _Christian G. Bower_, Jun 07 2005