cp's OEIS Frontend

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

%I A339158 #12 Nov 28 2020 19:54:58
%S A339158 1,1,2,4,8,18,37,84,180,413,902,2084,4628,10726,24128,56085,127421,
%T A339158 296955,680092,1588665,3662439,8574262,19875081,46628789,108584460,
%U A339158 255264307,596774173,1405626896,3297314994,7780687159,18305763571,43271547808,102069399803
%N A339158 Number of essentially parallel achiral series-parallel networks with n elements.
%C A339158 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.
%F A339158 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.
%e A339158 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'.
%e A339158 a(1) = 1: (o).
%e A339158 a(2) = 1: (o|o).
%e A339158 a(3) = 2: (o|oo).
%e A339158 a(4) = 4: (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).
%e A339158 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).
%e A339158 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):
%e A339158                A
%e A339158              /   \\
%e A339158             o      o   --- No reflective symmetry ---
%e A339158              \\  /
%e A339158                Z
%o A339158 (PARI) \\ here B(n) gives A003430 as a power series.
%o A339158 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A339158 B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
%o A339158 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))}
%Y A339158 Cf. A003430, A007454 (oriented), A339157, A339159, A339224 (unoriented).
%K A339158 nonn
%O A339158 1,3
%A A339158 _Andrew Howroyd_, Nov 27 2020