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%I #8 Nov 28 2020 19:54:46
%S 1,1,1,3,4,11,17,46,78,203,372,946,1830,4561,9207,22609,47166,114514,
%T 245154,590345,1289950,3087959,6858746,16352074,36800928,87502317,
%U 199036637,472483088,1084108363,2571356964,5942191918,14090541799,32754720101,77684033014,181473276607
%N Number of essentially series 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 configurations with n unit elements that are invariant under the reversal of all contained series configurations.
%F G.f.: x + (1 + P(x))*B(x^2) where P(x) is the g.f. of A339158 and B(x) is the g.f. of A003430.
%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: (oo).
%e a(3) = 1: (ooo).
%e a(4) = 3: (oooo), ((o|o)(o|o)), (o(o|o)o).
%e a(5) = 4: (ooooo), ((o|o)o(o|o)), (o(o|oo)o), (o(o|o|o)o).
%e a(6) = 11: (oooooo), ((o|o)oo(o|o)), (o(o|o)(o|o)o), ((o|oo)(o|oo)), ((o|o|o)(o|o|o)), (oo(o|o)oo), ((o|o)(o|o)(o|o)), (o(o|ooo)o), (o(oo|oo)o), (o(o|o|oo)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)); for(n=1, n\2, p = x + q*(1 + x + x*Ser(EulerT(Vec(p+(s-subst(p,x,x^2))/2))) - p)); Vec(p+O(x*x^n))}
%Y Cf. A003430, A007453 (oriented), A339158, A339159, A339223 (unoriented).
%K nonn
%O 1,4
%A _Andrew Howroyd_, Nov 27 2020
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