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%I #23 Sep 04 2023 23:38:35
%S 0,0,1,1,2,4,9,20,47,112,274,678,1709,4346,11176,28966,75656,198814,
%T 525496,1395758,3723986,9975314,26817655,72332320,195679137,530814386,
%U 1443556739,3934880554,10748839215,29420919456,80678144437,221618678694
%N Essentially series series-parallel networks with n unlabeled edges, multiple edges not allowed.
%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 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 r_n).
%H <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%F G.f. satisfies A(x) = A058385(x) - x + x^2.
%t (* f = g.f. of A058385 *) max = 31; f[x_] := Sum[b[n]*x^n, {n, 0, max}]; b[0] = 0; b[1] = 1; b[2] = 0; b[3] = 1; coef = CoefficientList[ Series[1 - x + x^2 + 2*f[x] - Product[(1 - x^j)^(-b[j]), {j, 1, max}], {x, 0, max}], x][[ 5 ;; All]]; g[x_] := Sum[a[n]*x^n, {n, 0, max}]; a[0] = a[1] = 0; a[2] = a[3] = 1; coeg = CoefficientList[ Series[g[x] - f[x] + x - x^2, {x, 0, max}], x][[ 5 ;; All]]; solf = SolveAlways[ Thread[coef == 0], x] ; solg = SolveAlways[ Thread[coeg == 0] /. solf[[1]], x]; Table[a[n], {n, 0, max}] /. solg[[1]] (* _Jean-François Alcover_, Jul 18 2012 *)
%t terms = 32; (* f = g.f. of A058385 *) f[_] = 0; Do[f[x_] = (1/2)*(-1 + x - x^2 + Product[(1 - x^j)^(-Ceiling[Coefficient[f[x], x, j]]), {j, 1, terms}]) + O[x]^ terms // Normal, 4*terms]; A[x_] = f[x] - x + x^2 + O[x]^terms; CoefficientList[A[x], x] (* _Jean-François Alcover_, Jan 10 2018 *)
%Y Cf. A058379, A058385, A058387.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_, Dec 20 2000
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