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A339301
Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.
4
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.
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
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Dec 22 2020
STATUS
approved