A339158 Number of essentially parallel achiral series-parallel networks with n elements.

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

Offset: 1

Author

Andrew Howroyd, Nov 27 2020

Keywords

• nonn

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.

Examples

			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
		

Crossrefs

Cf. A003430, A007454 (oriented), A339157, A339159, A339224 (unoriented).

Programs

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))}

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.