A058381 Number of series-parallel networks with n labeled edges, multiple edges not allowed.
0, 1, 1, 4, 20, 156, 1472, 17396, 239612, 3827816, 69071272, 1394315088, 31081310944, 758901184432, 20135117147056, 576927779925568, 17752780676186432, 583910574851160000, 20443098012485430272, 759064322969950283072, 29793617955495321025472
Offset: 0
Links
- Steven R. Finch, Series-parallel networks
- Steven R. Finch, Series-parallel networks, July 7, 2003. [Cached copy, with permission of the author]
- J. 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)
Programs
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Mathematica
max=19; f[x_] := -2*ProductLog[-Sqrt[1+x]/(2*Sqrt[E])]-1; CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, May 21 2012, after Vladeta Jovovic *) -
Maxima
a(n):=sum((sum((m+k-1)!*sum(((-1)^j*sum((2^(j-l)*(-1)^l *stirling1(m-l+j-1,j-l))/(l!*(m-l+j-1)!),l,0,j))/(k-j)!,j,0,k),k,0,m-1)) *stirling1(n,m),m,1,n); /* Vladimir Kruchinin, Feb 17 2012 */
Formula
E.g.f.: -2*LambertW(-1/2*exp(-1/2)*(1+x)^(1/2))-1. - Vladeta Jovovic, Aug 21 2006
a(n) = Sum(m=1..n, (Sum(k=0..m-1, (m+k-1)!*Sum(j=0..k, ((-1)^j *Sum(L=0..j, (2^(j-l)*(-1)^L*Stirling1(m-L+j-1,j-L))/(l!*(m-L+j-1)!)))/(k-j)!)))*Stirling1(n,m)). - Vladimir Kruchinin, Feb 17 2012
a(n) ~ n^(n-1) / (sqrt(2) * (4-exp(1))^(n-1/2)). - Vaclav Kotesovec, Jul 09 2013
a(n) = Sum_{k=1..n} Stirling1(n, k) * A006351(k), n > 0. - Sean A. Irvine, Feb 03 2018