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

Author

Andrew Howroyd, Nov 26 2020

Keywords

• nonn

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.

Examples

			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).
		

Crossrefs

Cf. A003430, A339151, A339155, A339156.

Programs

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)}

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.