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A339159 Number of achiral series-parallel networks with n elements. 5

%I

%S 1,2,3,7,12,29,54,130,258,616,1274,3030,6458,15287,33335,78694,174587,

%T 411469,925246,2179010,4952389,11662221,26733827,62980863,145385388,

%U 342766624,795810810,1878109984,4381423357,10352044123,24247955489,57362089607

%N Number of 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 series or parallel configurations with n unit elements that are invariant under the reversal of all contained series configurations.

%F a(n) = A339157(n) + A339158(n) for n > 1.

%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) = 2: (oo), (o|o).

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

%e a(4) = 7: (oooo), ((o|o)(o|o)), (o(o|o)o).

%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), t=p); for(n=1, n\2, t=x + q*(1 + p); p=x + x*Ser(EulerT(Vec(t+(s-subst(t,x,x^2))/2))) - t); Vec(p+t-x+O(x*x^n))}

%Y Cf. A003430 (oriented), A339157, A339158, A339225 (unoriented).

%K nonn

%O 1,2

%A _Andrew Howroyd_, Nov 27 2020

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Last modified October 6 16:34 EDT 2022. Contains 357270 sequences. (Running on oeis4.)