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%I #37 Apr 14 2019 11:08:25
%S 0,1,1,4,20,156,1472,17396,239612,3827816,69071272,1394315088,
%T 31081310944,758901184432,20135117147056,576927779925568,
%U 17752780676186432,583910574851160000,20443098012485430272,759064322969950283072,29793617955495321025472
%N Number of series-parallel networks with n labeled edges, multiple edges not allowed.
%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Series-parallel networks</a>
%H S. R. Finch, <a href="/A000084/a000084_2.pdf">Series-parallel networks</a>, July 7, 2003. [Cached copy, with permission of the author]
%H J. W. Moon, <a href="http://dx.doi.org/10.1016/S0304-0208(08)73057-3">Some enumerative results on series-parallel networks</a>, Annals Discrete Math., 33 (1987), 199-226 (the sequence V_n).
%H <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%F E.g.f.: -2*LambertW(-1/2*exp(-1/2)*(1+x)^(1/2))-1. - _Vladeta Jovovic_, Aug 21 2006
%F 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
%F a(n) ~ n^(n-1) / (sqrt(2) * (4-exp(1))^(n-1/2)). - _Vaclav Kotesovec_, Jul 09 2013
%F a(n) = Sum_{k=1..n} Stirling1(n, k) * A006351(k), n > 0. - _Sean A. Irvine_, Feb 03 2018
%t max=19; f[x_] := -2*ProductLog[-Sqrt[1+x]/(2*Sqrt[E])]-1;
%t CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* _Jean-François Alcover_, May 21 2012, after _Vladeta Jovovic_ *)
%o (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 */
%Y Equals A058379 + A058380.
%Y Cf. A006351.
%K nonn,nice,easy
%O 0,4
%A _N. J. A. Sloane_, Dec 19 2000
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