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A058385 Number of essentially parallel series-parallel networks with n unlabeled edges, multiple edges not allowed. 3
0, 1, 0, 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 (list; graph; refs; listen; history; text; internal format)
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]

Ji Li, Combinatorial Logarithm and Point-Determining Cographs, Electronic Journal of Combinatorics, 19 (3) (2012), #P8.

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

Index entries for sequences mentioned in Moon (1987)

FORMULA

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

MAPLE

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);

MATHEMATICA

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. *)

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 *)

CROSSREFS

Cf. A058379, A058386, A058387.

Sequence in context: A196244 A035084 A213905 * A058386 A095980 A036619

Adjacent sequences:  A058382 A058383 A058384 * A058386 A058387 A058388

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Dec 20 2000

STATUS

approved

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Last modified January 17 20:36 EST 2020. Contains 330987 sequences. (Running on oeis4.)