

A339158


Number of essentially parallel achiral seriesparallel networks with n elements.


4



1, 1, 2, 4, 8, 18, 37, 84, 180, 413, 902, 2084, 4628, 10726, 24128, 56085, 127421, 296955, 680092, 1588665, 3662439, 8574262, 19875081, 46628789, 108584460, 255264307, 596774173, 1405626896, 3297314994, 7780687159, 18305763571, 43271547808, 102069399803
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.


LINKS



FORMULA

G.f.: x  S(x)  1 + exp(Sum_{k>=1} (S(x^k) + (R(x^(2*k))  S(x^(2*k)))/2)/k) where S(x) is the g.f. of A339157 and R(x) is the g.f. of A007453.


EXAMPLE

In the following examples of seriesparallel 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: (oo).
a(3) = 2: (ooo).
a(4) = 4: (oooo), (oooo), (oooo), (oooo).
a(5) = 8: (ooooo), (o(oo)(oo)), (oo(oo)o), (ooooo), (ooooo), (ooooo), (ooooo), (ooooo).
a(6) = 16 includes (o(oo)(oo)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+(ssubst(t, x, x^2))/2)))  t); Vec(p+O(x*x^n))}


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



