A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

0, 1, 1, 2, 4, 8, 18, 40, 94, 224, 548, 1356, 3418, 8692, 22352, 57932, 151312, 397628, 1050992, 2791516, 7447972, 19950628, 53635310, 144664640, 391358274, 1061628772, 2887113478, 7869761108, 21497678430, 58841838912, 161356288874

Offset: 0

Author

N. J. A. Sloane, Dec 20 2000

Keywords

• nonn
• nice
• easy

Comments

This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel. See A000084 for examples.

Order is not considered significant in series configurations. - Andrew Howroyd, Dec 22 2020

Examples

			From _Andrew Howroyd_, Dec 22 2020: (Start)
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element (an edge) is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)),  (oo|ooo), (o|oo|oo).
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Steven R. Finch, Series-parallel networks
• Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
• John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).
• Index entries for sequences mentioned in Moon (1987)

Crossrefs

A000084 is the case that multiple edges are allowed.

A058381 is the case that edges are labeled.

A339290 is the case that order is significant in series configurations.

Cf. A058385, A058386, A000311, A000669, A006351.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(s=p=vector(n)); p[1]=1; for(n=2, n, s[n]=EulerT(p[1..n])[n]; p[n]=vecsum(EulerT(s[1..n])[n-1..n])-s[n]); concat([0], p+s)} \\ Andrew Howroyd, Dec 22 2020

Formula

a(n) = A058385(n) + A058386(n).