A108529 Number of asymmetric mobiles (cycle rooted trees) with n generators.

1, 1, 2, 5, 16, 51, 177, 621, 2246, 8245, 30783, 116257, 443945, 1710255, 6640939, 25961690, 102105115, 403701135, 1603721999, 6397931901, 25621989760, 102965680728, 415091909292, 1678226164646, 6803121058354, 27645628327636

Offset: 1

Author

Christian G. Bower, Jun 07 2005

Keywords

• nonn

Comments

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

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• C. G. Bower, Transforms (2)
• Index entries for sequences related to mobiles

Crossrefs

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

Programs

PARI

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

Formula

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

From Petros Hadjicostas, Dec 05 2017: (Start)

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.

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.)

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 + ...

(End)