A339290
Number of oriented series-parallel networks with n elements and without multiple unit elements in parallel.
Original entry on oeis.org
1, 1, 2, 5, 13, 36, 103, 306, 930, 2887, 9100, 29082, 93951, 306414, 1007361, 3335088, 11108986, 37203873, 125193694, 423099557, 1435427202, 4886975378, 16690971648, 57172387872, 196358421066, 676050576441, 2332887221847, 8067160995797, 27950871439353, 97019613539949
Offset: 1
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 5: (oooo), (o(o|oo)), ((o|oo)o), (o|ooo), (oo|oo).
a(5) = 13: (ooooo), (oo(o|oo)), (o(o|oo)o), ((o|oo)oo), (o(o|ooo)), (o(oo|oo)), ((o|ooo)o), ((oo|oo)o), (o|oooo), (o|o(o|oo)), (o|(o|oo)o), (oo|ooo), (o|oo|oo).
A003430 is the case with multiple unit elements in parallel allowed.
A058387 is the case that order is not 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, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); Vec(p)}
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 *)
A058386
Essentially series series-parallel networks with n unlabeled edges, multiple edges not allowed.
Original entry on oeis.org
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
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(* 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 *)
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