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 A339158 Number of essentially parallel achiral series-parallel networks with n elements. 4

%I #12 Nov 28 2020 19:54:58

%S 1,1,2,4,8,18,37,84,180,413,902,2084,4628,10726,24128,56085,127421,

%T 296955,680092,1588665,3662439,8574262,19875081,46628789,108584460,

%U 255264307,596774173,1405626896,3297314994,7780687159,18305763571,43271547808,102069399803

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

%C 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 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 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 a(1) = 1: (o).

%e a(2) = 1: (o|o).

%e a(3) = 2: (o|oo).

%e a(4) = 4: (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).

%e 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 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 A

%e / \\

%e o o --- No reflective symmetry ---

%e \\ /

%e Z

%o (PARI) \\ here B(n) gives A003430 as a power series.

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}

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

%K nonn

%O 1,3

%A _Andrew Howroyd_, Nov 27 2020

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Last modified August 11 11:30 EDT 2024. Contains 375068 sequences. (Running on oeis4.)