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%I A339301 #8 Dec 22 2020 18:26:34
%S A339301 1,2,12,108,1380,22440,446040,10461360,282970800,8670594240,
%T A339301 296850597120,11230473925440,465262142304960,20948652798353280,
%U A339301 1018583225567107200,53190962586022060800,2969038807022050963200,176410305542414738995200,11116489894884127122969600
%N A339301 Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.
%C A339301 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 A339301 a(n) = A339299(n) + A339300(n).
%F A339301 E.g.f.: A(x) satisfies A(x) = (1 + x)*exp(A(x)^2/(1+A(x))) - 1.
%F A339301 E.g.f.: P(x)/(1 - P(x)) where P(x) is the e.g.f. of A339300.
%F A339301 E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048172.
%e A339301 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 A339301 (PARI) \\ Note giving Z=exp(x)-1 gives A048172.
%o A339301 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 A339301 (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 A339301 A048172 is the case with multiple unit elements in parallel allowed.
%Y A339301 A058381 is the case that order is not significant in series configurations.
%Y A339301 Main diagonal of A339297.
%Y A339301 Cf. A339290 (unlabeled), A339299, A339300.
%K A339301 nonn
%O A339301 1,2
%A A339301 _Andrew Howroyd_, Dec 22 2020

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