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

0,4

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

References

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

Index entries for sequences mentioned in Moon (1987)

S. R. Finch, Series-parallel networks

S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]

Formula

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

Example

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)

Prog

(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

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.

Sequence in context: A000967 A288309 A096813 * A330052 A317787 A019231

Adjacent sequences: A058384 A058385 A058386 * A058388 A058389 A058390

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Dec 20 2000

Status

approved