OFFSET
1,3
COMMENTS
A series configuration is the unit element or an ordered concatenation of two or more parallel configurations and a parallel configuration is the unit element or a multiset of two or more series configurations. a(n) is the number of parallel configurations with n unit elements that are invariant under the reversal of all contained series configurations.
FORMULA
EXAMPLE
In the following examples of series-parallel networks, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (o|o).
a(3) = 2: (o|oo).
a(4) = 4: (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).
a(5) = 8: (o|oooo), (o|(o|o)(o|o)), (o|o(o|o)o), (oo|ooo), (o|o|ooo), (o|oo|oo), (o|o|o|oo), (o|o|o|o|o).
a(6) = 16 includes (o(o|o)|(o|o)o) which is the first example of a network that is achiral but does not have reflective symmetry when embedded in the plane as shown below (edges correspond to elements):
A
/ \\
o o --- No reflective symmetry ---
\\ /
Z
PROG
(PARI) \\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, my(t=x + q*(1 + p)); p=x + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+O(x*x^n))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 27 2020
STATUS
approved