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A339156 Number of oriented series-parallel networks with n elements and without unit elements in parallel. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 12 23:52 EDT 2022. Contains 356077 sequences. (Running on oeis4.)