A058379 Essentially parallel series-parallel networks with n labeled edges, multiple edges not allowed.
0, 1, 0, 3, 7, 90, 676, 9058, 117286, 1934068, 34354196, 698971944, 15520697072, 379690093016, 10064445063128, 288507479108384, 8875736500909216, 291965748820524000, 10221371162528667136, 379535362671828005536, 14896748155197456096736
Offset: 0
Examples
A(x) = x + 1/2*x^3 + 7/24*x^4 + 3/4*x^5 + 169/180*x^6 + ... For n=4 there are two unlabeled networks: ..o.....o--o ./.\.../....\ o...o o------o .\./ ..o which can be labeled in 3 (resp. 4) ways, for a total of 7.
Links
- 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 Q_n).
- Index entries for sequences mentioned in Moon (1987)
Programs
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Maple
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 Order := 50; t1 := solve(series((exp(A)-2*A-1),A)=-log(1+x),A); A058379 := n-> n!*coeff(t1,x,n); # method 2
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Mathematica
CoefficientList[InverseSeries[Series[-1+E^(1+2*a-E^a), {a, 0, 20}], x], x]*Range[0, 20]! (* Jean-François Alcover, Jul 21 2011 *) 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 *) -
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 */
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PARI
Vec(serlaplace(-1/2 + log(1+x)/2 - lambertw(-exp(-.5)*sqrt(1+x)/2))) \/ 1 \\ Charles R Greathouse IV, Jun 16 2021
Formula
E.g.f. satisfies A(x) = x + O(x^2), 2*A(x) = exp(A(x)) - 1 + log(1+x).
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
E.g.f.: -1/2 + log(1+x)/2 - LambertW(-exp(-1/2)*sqrt(1+x)/2). - Vaclav Kotesovec, Jan 08 2014
a(n) ~ n^(n-1) / (2*sqrt(2) * (4-exp(1))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014