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

0,4

Links

Table of n, a(n) for n=0..20.

S. R. Finch, Series-parallel networks

S. 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)

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

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 *)

Prog

(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 */

Crossrefs

Equals A058379 + A058380.

Cf. A006351.

Sequence in context: A119022 A006682 A115852 * A094651 A065526 A032333

Adjacent sequences: A058378 A058379 A058380 * A058382 A058383 A058384

Keyword

nonn,nice,easy

Author

N. J. A. Sloane, Dec 19 2000

Status

approved