A339155 Number of essentially parallel oriented series-parallel networks with n elements and without unit elements in parallel.

1, 0, 0, 1, 1, 3, 5, 13, 29, 70, 165, 409, 1001, 2505, 6278, 15904, 40447, 103567, 266229, 687668, 1782573, 4637731, 12103112, 31679212, 83135973, 218713492, 576683119, 1523740365, 4033915677, 10698680606, 28422818782, 75629586540, 201539697208, 537818080714

Offset

1,6

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 number of parallel configurations with n unit elements.

Links

Table of n, a(n) for n=1..34.

Formula

G.f.: B(x)/(1 + B(x)) where B(x) is the g.f. of A339156.

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(4) = 1: (oo|oo).
a(5) = 1: (oo|ooo).
a(6) = 3: (oo|oooo), (ooo|ooo), (oo|oo|oo).
a(7) = 4: (oo|ooooo), (oo|o(oo|oo)), (oo|(oo|oo)o), (ooo|oooo), (oo|oo|ooo).

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(1 - 1/(1+p))}

Crossrefs

Cf. A003430, A339152, A339154, A339156.

Sequence in context: A290113 A238216 A067932 * A168314 A335562 A106666

Adjacent sequences: A339152 A339153 A339154 * A339156 A339157 A339158

Keyword

nonn

Author

Andrew Howroyd, Nov 26 2020

Status

approved