cp's OEIS Frontend

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

%I A339156 #10 Nov 28 2020 01:21:41
%S A339156 1,1,1,2,4,9,19,43,99,235,562,1370,3369,8380,21000,53038,134759,
%T A339156 344390,884376,2281274,5907791,15354795,40037979,104712010,274600650,
%U A339156 721931534,1902362100,5023654075,13292543205,35237009037,93570419556,248873359877,662940466647
%N A339156 Number of oriented series-parallel networks with n elements and without unit elements in parallel.
%C A339156 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 A339156 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 A339156 a(n) = A339154(n) + A339155(n).
%F A339156 Euler transform of A339154 gives this sequence with a(1) = 0.
%F A339156 G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339155.
%F A339156 G.f.: S(x)/2 + sqrt(S(x) + S(x)^2/4) where S(x) is the g.f. of A339154.
%e A339156 In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
%e A339156 a(1) = 1: (o)
%e A339156 a(2) = 1: (oo).
%e A339156 a(3) = 1: (ooo).
%e A339156 a(4) = 2: (oooo), (oo|oo).
%e A339156 a(5) = 4: (ooooo), (o(oo|oo)), ((oo|oo)o), (oo|ooo).
%e A339156 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 A339156 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A339156 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 A339156 Cf. A003430, A339153, A339154, A339155.
%K A339156 nonn
%O A339156 1,4
%A A339156 _Andrew Howroyd_, Nov 26 2020