A058387
Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.
Original entry on oeis.org
0, 1, 1, 2, 4, 8, 18, 40, 94, 224, 548, 1356, 3418, 8692, 22352, 57932, 151312, 397628, 1050992, 2791516, 7447972, 19950628, 53635310, 144664640, 391358274, 1061628772, 2887113478, 7869761108, 21497678430, 58841838912, 161356288874
Offset: 0
From _Andrew Howroyd_, Dec 22 2020: (Start)
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element (an edge) is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)), (oo|ooo), (o|oo|oo).
(End)
- Andrew Howroyd, Table of n, a(n) for n = 0..500
- Steven R. Finch, Series-parallel networks
- Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
- John W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence v_n).
- Index entries for sequences mentioned in Moon (1987)
A000084 is the case that multiple edges are allowed.
A058381 is the case that edges are labeled.
A339290 is the case that order is significant in series configurations.
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EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(s=p=vector(n)); p[1]=1; for(n=2, n, s[n]=EulerT(p[1..n])[n]; p[n]=vecsum(EulerT(s[1..n])[n-1..n])-s[n]); concat([0], p+s)} \\ Andrew Howroyd, Dec 22 2020
A058385
Number of essentially parallel series-parallel networks with n unlabeled edges, multiple edges not allowed.
Original entry on oeis.org
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
- 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)
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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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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 *)
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