%I #29 Jun 07 2021 13:02:17

%S 0,1,0,1,2,4,9,20,47,112,274,678,1709,4346,11176,28966,75656,198814,

%T 525496,1395758,3723986,9975314,26817655,72332320,195679137,530814386,

%U 1443556739,3934880554,10748839215,29420919456,80678144437,221618678694

%N Number of essentially parallel series-parallel networks with n unlabeled edges, multiple edges not allowed.

%H Vaclav Kotesovec, <a href="/A058385/b058385.txt">Table of n, a(n) for n = 0..500</a> (using data from A058387)

%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]

%H Ji Li, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p8">Combinatorial Logarithm and Point-Determining Cographs</a>, Electronic Journal of Combinatorics, 19 (3) (2012), #P8.

%H J. 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 q_n).

%H <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>

%F G.f. satisfies 1 - x + x^2 + 2*A(x) = Product_{j>=1} (1-x^j)^(-a(j)).

%p Q := x; q[1] := 1; for d from 1 to 40 do q[d+1] := c; Q := Q+c*x^(d+1); t0 := mul((1-x^j)^(-q[j]),j=1..d+1); t01 := series(t0,x,d+2); t05 := series(2*Q +1-x+x^2 -t01, x, d+2); t1 := coeff(t05,x,d+1); t2 := solve(t1,c); q[d+1] := t2; Q := subs(c=t2,Q); Q := series(Q,x,d+2); od: A058385 := n->coeff(Q,x,n);

%t max = 31; f[x_] := Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 0; a[3] = 1; se = Series[ 1 - x + x^2 + 2*f[x] - Product[(1 - x^j)^(-a[j]), {j, 1, max}], {x, 0, max}]; sol = Solve[ Thread[ CoefficientList[ se, x] == 0]]; A058385 = Table[a[n], {n, 0, max}] /. First[sol] (* _Jean-François Alcover_, Dec 27 2011, after g.f. *)

%t terms = 32; A[_] = 0; Do[A[x_] = (1/2)*(-1 + x - x^2 + Product[(1 - x^j)^(-Ceiling[Coefficient[A[x], x, j]]), {j, 1, terms}]) + O[x]^ terms // Normal, 4*terms]; CoefficientList[A[x] + O[x]^terms, x] (* _Jean-François Alcover_, Jan 10 2018 *)

%Y Cf. A058379, A058386, A058387.

%K nonn,easy,nice

%O 0,5

%A _N. J. A. Sloane_, Dec 20 2000