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%I #7 Nov 28 2020 19:55:18
%S 1,1,2,6,17,57,196,723,2729,10638,42161,169912,692703,2853523,
%T 11852644,49592966,208800209,883970867,3760605627,16068272965,
%U 68925340187,296705390322,1281351319402,5549911448062,24103086681839,104938476264310,457920147387969,2002462084788769
%N Number of essentially series unoriented series-parallel networks with n elements.
%C See A339225 for additional details.
%F a(n) = (A007453(n) + A339157(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: (ooo), (o(o|o)).
%e a(4) = 6: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)(o|o)), (o(o|oo)), (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+x+subst(x^2/(1+x),x,B(n)))/2}
%Y Cf. A003430, A007453 (oriented), A339157 (achiral), A339224, A339225.
%K nonn
%O 1,3
%A _Andrew Howroyd_, Nov 27 2020
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