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%I #8 Dec 22 2020 18:26:34
%S 1,2,12,108,1380,22440,446040,10461360,282970800,8670594240,
%T 296850597120,11230473925440,465262142304960,20948652798353280,
%U 1018583225567107200,53190962586022060800,2969038807022050963200,176410305542414738995200,11116489894884127122969600
%N Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.
%C 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.
%F a(n) = A339299(n) + A339300(n).
%F E.g.f.: A(x) satisfies A(x) = (1 + x)*exp(A(x)^2/(1+A(x))) - 1.
%F E.g.f.: P(x)/(1 - P(x)) where P(x) is the e.g.f. of A339300.
%F E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048172.
%e 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).
%o (PARI) \\ Note giving Z=exp(x)-1 gives A048172.
%o 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))}
%o (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))))}
%Y A048172 is the case with multiple unit elements in parallel allowed.
%Y A058381 is the case that order is not significant in series configurations.
%Y Main diagonal of A339297.
%Y Cf. A339290 (unlabeled), A339299, A339300.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Dec 22 2020
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