A339290 Number of oriented series-parallel networks with n elements and without multiple unit elements in parallel.

1, 1, 2, 5, 13, 36, 103, 306, 930, 2887, 9100, 29082, 93951, 306414, 1007361, 3335088, 11108986, 37203873, 125193694, 423099557, 1435427202, 4886975378, 16690971648, 57172387872, 196358421066, 676050576441, 2332887221847, 8067160995797, 27950871439353, 97019613539949

Offset

1,3

Comments

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. a(n) is the total number of series and parallel configurations with n unit elements.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..500

Formula

a(n) = A339288(n) + A339289(n).

G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339289.

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) = 2: (ooo), (o|oo).
a(4) = 5: (oooo), (o(o|oo)), ((o|oo)o), (o|ooo), (oo|oo).
a(5) = 13: (ooooo), (oo(o|oo)), (o(o|oo)o), ((o|oo)oo), (o(o|ooo)), (o(oo|oo)), ((o|ooo)o), ((oo|oo)o), (o|oooo), (o|o(o|oo)), (o|(o|oo)o), (oo|ooo), (o|oo|oo).

Prog

(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); Vec(p)}

Crossrefs

A003430 is the case with multiple unit elements in parallel allowed.

A058387 is the case that order is not significant in series configurations.

Cf. A339156, A339288, A339289, A339293 (achiral), A339296 (unoriented), A339301 (labeled).

Sequence in context: A099164 A358460 A289453 * A036765 A246555 A366023

Adjacent sequences: A339287 A339288 A339289 * A339291 A339292 A339293

Keyword

nonn

Author

Andrew Howroyd, Dec 07 2020

Status

approved