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

1,4

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.

Links

Table of n, a(n) for n=1..33.

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.

Example

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).

Prog

(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)}

Crossrefs

Cf. A003430, A339153, A339154, A339155.

Sequence in context: A206301 A026776 A117160 * A247623 A084083 A036611

Adjacent sequences: A339153 A339154 A339155 * A339157 A339158 A339159

Keyword

nonn

Author

Andrew Howroyd, Nov 26 2020

Status

approved