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%I #33 Jun 16 2021 13:44:26
%S 0,1,0,3,7,90,676,9058,117286,1934068,34354196,698971944,15520697072,
%T 379690093016,10064445063128,288507479108384,8875736500909216,
%U 291965748820524000,10221371162528667136,379535362671828005536,14896748155197456096736
%N Essentially parallel series-parallel networks with n labeled edges, multiple edges not allowed.
%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="https://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 Q_n).
%H <a href="/index/Mo#Moon87">Index entries for sequences mentioned in Moon (1987)</a>
%F E.g.f. satisfies A(x) = x + O(x^2), 2*A(x) = exp(A(x)) - 1 + log(1+x).
%F a(n) = sum(m=1..n, (sum(k=0..m-1, (m+k-1)!*sum(j=0..k, sum(i=0..j, ((-1)^i*2^i*Stirling2(m+j-i-1,j-i))/(i!*(m+j-i-1)!))/(k-j)!))) *Stirling1(n,m)). - _Vladimir Kruchinin_, Feb 17 2012
%F E.g.f.: -1/2 + log(1+x)/2 - LambertW(-exp(-1/2)*sqrt(1+x)/2). - _Vaclav Kotesovec_, Jan 08 2014
%F a(n) ~ n^(n-1) / (2*sqrt(2) * (4-exp(1))^(n-1/2)). - _Vaclav Kotesovec_, Jan 08 2014
%e A(x) = x + 1/2*x^3 + 7/24*x^4 + 3/4*x^5 + 169/180*x^6 + ...
%e For n=4 there are two unlabeled networks:
%e ..o.....o--o
%e ./.\.../....\
%e o...o o------o
%e .\./
%e ..o
%e which can be labeled in 3 (resp. 4) ways, for a total of 7.
%p Q := x; for d from 1 to 30 do Q := Q+c*x^(d+1)/(d+1)!; t1 := coeff(series(2*Q - (exp(Q)-1+log(1+x)), x, d+2), x, d+1); t2 := solve(t1,c); Q := subs(c=t2,Q); Q := series(Q,x,d+2); od: A058379 := n->coeff(Q,x,n)*n!; # method 1
%p Order := 50; t1 := solve(series((exp(A)-2*A-1),A)=-log(1+x),A); A058379 := n-> n!*coeff(t1,x,n); # method 2
%t CoefficientList[InverseSeries[Series[-1+E^(1+2*a-E^a), {a, 0, 20}], x], x]*Range[0, 20]! (* _Jean-François Alcover_, Jul 21 2011 *)
%t CoefficientList[Series[(-1 + Log[1+x] - 2*ProductLog[-Sqrt[1+x]/(2*Sqrt[E])])/2,{x,0,15}],x] * Range[0,15]! (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o (Maxima) a(n):=sum((sum((m+k-1)!*sum(sum(((-1)^i*2^i*stirling2(m+j-i-1, j-i))/(i!*(m+j-i-1)!),i,0,j)/(k-j)!,j,0,k),k,0,m-1)) *stirling1(n,m), m,1,n); /* _Vladimir Kruchinin_, Feb 17 2012 */
%o (PARI) Vec(serlaplace(-1/2 + log(1+x)/2 - lambertw(-exp(-.5)*sqrt(1+x)/2))) \/ 1 \\ _Charles R Greathouse IV_, Jun 16 2021
%Y Cf. A058380, A058381. See A000669 for unlabeled case when parallel edges are allowed.
%K nonn,easy,nice
%O 0,4
%A _N. J. A. Sloane_, Dec 19 2000
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