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

1, 1, 1, 2, 4, 9, 19, 43, 99, 235, 562, 1370, 3369, 8380, 21000, 53038, 134759, 344390, 884376, 2281274, 5907791, 15354795, 40037979, 104712010, 274600650, 721931534, 1902362100, 5023654075, 13292543205, 35237009037, 93570419556, 248873359877, 662940466647

Offset: 1

Author

Andrew Howroyd, Nov 26 2020

Keywords

• nonn

Comments

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.

Examples

			In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o)
a(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 2: (oooo), (oo|oo).
a(5) = 4: (ooooo), (o(oo|oo)), ((oo|oo)o), (oo|ooo).
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).
		

Crossrefs

Cf. A003430, A339153, A339154, A339155.

Programs

PARI

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

Formula

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).

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

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

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

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