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
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..500 (using data from A058387)
- Steven R. Finch, Series-parallel networks
- Steven 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.
- John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence q_n).
- Wei Wang and Ximei Huang, Almost all cographs have a cospectral mate, arXiv:2507.16730 [math.CO], 2025. See p. 6.
- Index entries for sequences mentioned in Moon (1987)
Programs
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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);
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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 *)
Formula
G.f. satisfies 1 - x + x^2 + 2*A(x) = Product_{j>=1} (1-x^j)^(-a(j)).