|
%I #16 Dec 22 2020 14:32:04
%S 0,1,1,2,4,8,18,40,94,224,548,1356,3418,8692,22352,57932,151312,
%T 397628,1050992,2791516,7447972,19950628,53635310,144664640,391358274,
%U 1061628772,2887113478,7869761108,21497678430,58841838912,161356288874
%N Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.
%C 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 Order is not considered significant in series configurations. - _Andrew Howroyd_, Dec 22 2020
%D J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).
%H Andrew Howroyd, <a href="/A058387/b058387.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Series-parallel networks</a>
%H S. R. Finch, <a href="/A000084/a000084_2.pdf">Series-parallel networks</a>, July 7, 2003. [Cached copy, with permission of the author]
%F a(n) = A058385(n) + A058386(n).
%e From _Andrew Howroyd_, Dec 22 2020: (Start)
%e 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 a(1) = 1: (o).
%e a(2) = 1: (oo).
%e a(3) = 2: (ooo), (o|oo).
%e a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
%e 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 (End)
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o 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 A000084 is the case that multiple edges are allowed.
%Y A058381 is the case that edges are labeled.
%Y A339290 is the case that order is significant in series configurations.
%Y Cf. A058385, A058386, A000311, A000669, A006351.
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_, Dec 20 2000
|
