A339154 Number of essentially series oriented series-parallel networks with n elements and without unit elements in parallel.

0, 1, 1, 1, 3, 6, 14, 30, 70, 165, 397, 961, 2368, 5875, 14722, 37134, 94312, 240823, 618147, 1593606, 4125218, 10717064, 27934867, 73032798, 191464677, 503218042, 1325678981, 3499913710, 9258627528, 24538328431, 65147600774, 173243773337, 461400769439

Offset

1,5

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 series configurations with n unit elements.

Links

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

Formula

G.f.: P(x)^2/(1 - P(x)) where P(x) is the g.f. of A339155.

G.f.: B(x)^2/(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(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 1: (oooo).
a(5) = 3: (ooooo), (o(oo|oo)), ((oo|oo)o).
a(6) = 6: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o).

Prog

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

Crossrefs

Cf. A003430, A339151, A339155, A339156.

Sequence in context: A131244 A077926 A091601 * A063119 A218982 A106803

Adjacent sequences: A339151 A339152 A339153 * A339155 A339156 A339157

Keyword

nonn

Author

Andrew Howroyd, Nov 26 2020

Status

approved