A339301 Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

1, 2, 12, 108, 1380, 22440, 446040, 10461360, 282970800, 8670594240, 296850597120, 11230473925440, 465262142304960, 20948652798353280, 1018583225567107200, 53190962586022060800, 2969038807022050963200, 176410305542414738995200, 11116489894884127122969600

Offset

1,2

Comments

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. a(n) is the total number of series and parallel configurations with n unit elements labeled 1..n.

Links

Table of n, a(n) for n=1..19.

Formula

a(n) = A339299(n) + A339300(n).

E.g.f.: A(x) satisfies A(x) = (1 + x)*exp(A(x)^2/(1+A(x))) - 1.

E.g.f.: P(x)/(1 - P(x)) where P(x) is the e.g.f. of A339300.

E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048172.

Example

a(3) = 12 because there are 2 unlabeled structures each of which can be labeled in 6 ways. The unlabeled structures are (ooo) and (o|oo).

Prog

(PARI) \\ Note giving Z=exp(x)-1 gives A048172.
seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(p))}
(PARI) seq(n)={my(A=O(x*x^n)); Vec(serlaplace(subst(serreverse(log(1+x+A) - x^2/(1+x)), x, log(1+x+A))))}

Crossrefs

A048172 is the case with multiple unit elements in parallel allowed.

A058381 is the case that order is not significant in series configurations.

Main diagonal of A339297.

Cf. A339290 (unlabeled), A339299, A339300.

Sequence in context: A228173 A218652 A194786 * A179493 A193268 A235601

Adjacent sequences: A339298 A339299 A339300 * A339302 A339303 A339304

Keyword

nonn

Author

Andrew Howroyd, Dec 22 2020

Status

approved