A058386 Essentially series series-parallel networks with n unlabeled edges, multiple edges not allowed.

0, 0, 1, 1, 2, 4, 9, 20, 47, 112, 274, 678, 1709, 4346, 11176, 28966, 75656, 198814, 525496, 1395758, 3723986, 9975314, 26817655, 72332320, 195679137, 530814386, 1443556739, 3934880554, 10748839215, 29420919456, 80678144437, 221618678694

Offset

0,5

Links

Table of n, a(n) for n=0..31.

S. R. Finch, Series-parallel networks.

S. R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]

J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence r_n).

Index entries for sequences mentioned in Moon (1987)

Formula

G.f. satisfies A(x) = A058385(x) - x + x^2.

Mathematica

(* 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 *)
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 *)

Crossrefs

Cf. A058379, A058385, A058387.

Sequence in context: A363557 A213905 A058385 * A095980 A036619 A036620

Adjacent sequences: A058383 A058384 A058385 * A058387 A058388 A058389

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Dec 20 2000

Status

approved