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%I #10 Nov 28 2020 19:55:32
%S 1,2,4,11,30,98,328,1193,4459,17287,68283,274726,1118960,4607578,
%T 19135274,80063095,337104367,1427274619,6072510001,25949049372,
%U 111319539096,479243000380,2069825207344,8965693829582,38940393808337,169546919220357,739895248735963
%N Number of unoriented 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 distinct series or parallel configurations with n unit elements modulo reversing the order of all series configurations.
%F a(n) = (A003430(n) + A339159(n))/2.
%F a(n) = A339223(n) + A339224(n) for n > 1.
%F A000084(n) <= a(n) <= A003430(n).
%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) = 4: (ooo), (o(o|o)), (o|o|o), (o|oo).
%e a(4) = 11: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)), (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (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+B(n))/2}
%Y Cf. A000084, A003430 (oriented), A339159 (achiral), A339223, A339224.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Nov 27 2020
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