A339225 Number of unoriented series-parallel networks with n elements.

1, 2, 4, 11, 30, 98, 328, 1193, 4459, 17287, 68283, 274726, 1118960, 4607578, 19135274, 80063095, 337104367, 1427274619, 6072510001, 25949049372, 111319539096, 479243000380, 2069825207344, 8965693829582, 38940393808337, 169546919220357, 739895248735963

Offset: 1

Author

Andrew Howroyd, Nov 27 2020

Keywords

• nonn

Comments

A series configuration is the unit element or 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 distinct series or parallel configurations with n unit elements modulo reversing the order of all series configurations.

Examples

			In the following examples of series-parallel networks, 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) = 2: (oo), (o|o).
a(3) = 4: (ooo), (o(o|o)), (o|o|o), (o|oo).
a(4) = 11: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)),  (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (o|o(o|o)).
		

Crossrefs

Cf. A000084, A003430 (oriented), A339159 (achiral), A339223, A339224.

Programs

PARI

\\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2), t=p); for(n=1, n\2, t=x + q*(1 + p); p=x + x*Ser(EulerT(Vec(t+(s-subst(t,x,x^2))/2))) - t); Vec(p+t-x+B(n))/2}

Formula

a(n) = (A003430(n) + A339159(n))/2.

a(n) = A339223(n) + A339224(n) for n > 1.

A000084(n) <= a(n) <= A003430(n).