A058385 Number of essentially parallel series-parallel networks with n unlabeled edges, multiple edges not allowed.

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

Offset

0,5

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (using data from A058387)

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: A035084 A363557 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