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A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

%I A058387 #22 Dec 16 2024 13:28:14
%S A058387 0,1,1,2,4,8,18,40,94,224,548,1356,3418,8692,22352,57932,151312,
%T A058387 397628,1050992,2791516,7447972,19950628,53635310,144664640,391358274,
%U A058387 1061628772,2887113478,7869761108,21497678430,58841838912,161356288874
%N A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.
%C A058387 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.
%C A058387 Order is not considered significant in series configurations. - _Andrew Howroyd_, Dec 22 2020
%H A058387 Andrew Howroyd, <a href="/A058387/b058387.txt">Table of n, a(n) for n = 0..500</a>
%H A058387 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Series-parallel networks</a>
%H A058387 Steven R. Finch, <a href="/A000084/a000084_2.pdf">Series-parallel networks</a>, July 7, 2003. [Cached copy, with permission of the author]
%H A058387 John W. Moon, <a href="http://dx.doi.org/10.1016/S0304-0208(08)73057-3">Some enumerative results on series-parallel networks</a>, Annals Discrete Math., 33 (1987), 199-226  (the sequence v_n).
%H A058387 <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%F A058387 a(n) = A058385(n) + A058386(n).
%e A058387 From _Andrew Howroyd_, Dec 22 2020: (Start)
%e A058387 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'.
%e A058387 a(1) = 1: (o).
%e A058387 a(2) = 1: (oo).
%e A058387 a(3) = 2: (ooo), (o|oo).
%e A058387 a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
%e A058387 a(5) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)),  (oo|ooo), (o|oo|oo).
%e A058387 (End)
%o A058387 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A058387 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
%Y A058387 A000084 is the case that multiple edges are allowed.
%Y A058387 A058381 is the case that edges are labeled.
%Y A058387 A339290 is the case that order is significant in series configurations.
%Y A058387 Cf. A058385, A058386, A000311, A000669, A006351.
%K A058387 nonn,nice,easy
%O A058387 0,4
%A A058387 _N. J. A. Sloane_, Dec 20 2000