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A058381 Number of series-parallel networks with n labeled edges, multiple edges not allowed. 7
0, 1, 1, 4, 20, 156, 1472, 17396, 239612, 3827816, 69071272, 1394315088, 31081310944, 758901184432, 20135117147056, 576927779925568, 17752780676186432, 583910574851160000, 20443098012485430272, 759064322969950283072, 29793617955495321025472 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 9 02:22 EST 2022. Contains 358698 sequences. (Running on oeis4.)