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A339156 Number of oriented series-parallel networks with n elements and without unit elements in parallel. 4

%I

%S 1,1,1,2,4,9,19,43,99,235,562,1370,3369,8380,21000,53038,134759,

%T 344390,884376,2281274,5907791,15354795,40037979,104712010,274600650,

%U 721931534,1902362100,5023654075,13292543205,35237009037,93570419556,248873359877,662940466647

%N Number of oriented series-parallel networks with n elements and without unit elements in parallel.

%C A series configuration is 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 total number of series and parallel configurations with n unit elements.

%F G.f.: A(x) where A(x) satisfies A(x) = x - 1 + exp(Sum_{k>=1} (A(x^k) + 1/(1 + A(x^k)) - 1)/k).

%F a(n) = A339154(n) + A339155(n).

%F Euler transform of A339154 gives this sequence with a(1) = 0.

%F G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339155.

%F G.f.: S(x)/2 + sqrt(S(x) + S(x)^2/4) where S(x) is the g.f. of A339154.

%e In the following examples, 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) = 2: (oooo), (oo|oo).

%e a(5) = 4: (ooooo), (o(oo|oo)), ((oo|oo)o), (oo|ooo).

%e a(6) = 9: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o), (oo|oooo), (ooo|ooo), (oo|oo).

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n)={my(p=x+O(x^2)); for(n=2, n, p=x+x*Ser(EulerT(Vec(p^2/(1+p), -n)))); Vec(p)}

%Y Cf. A003430, A339153, A339154, A339155.

%K nonn

%O 1,4

%A _Andrew Howroyd_, Nov 26 2020

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Last modified October 3 04:21 EDT 2022. Contains 357230 sequences. (Running on oeis4.)