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%I #6 Nov 28 2020 19:55:25
%S 1,1,2,5,13,41,132,470,1730,6649,26122,104814,426257,1754055,7282630,
%T 30470129,128304158,543303752,2311904374,9880776407,42394198909,
%U 182537610058,788473887942,3415782381520,14837307126498,64608442956047,281975101347994,1233237605651194
%N Number of essentially parallel unoriented series-parallel networks with n elements.
%C See A339225 for additional details.
%F a(n) = (A007454(n) + A339158(n))/2.
%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), (o|o).
%e a(3) = 2: (o|o|o), (o|oo).
%e a(4) = 5: (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)); 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+subst(x/(1+x), x, B(n)))/2}
%Y Cf. A003430, A007454 (oriented), A339158 (achiral), A339223, A339225.
%K nonn
%O 1,3
%A _Andrew Howroyd_, Nov 27 2020
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