A058379
Essentially parallel series-parallel networks with n labeled edges, multiple edges not allowed.
Original entry on oeis.org
0, 1, 0, 3, 7, 90, 676, 9058, 117286, 1934068, 34354196, 698971944, 15520697072, 379690093016, 10064445063128, 288507479108384, 8875736500909216, 291965748820524000, 10221371162528667136, 379535362671828005536, 14896748155197456096736
Offset: 0
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.
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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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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 *)
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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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Vec(serlaplace(-1/2 + log(1+x)/2 - lambertw(-exp(-.5)*sqrt(1+x)/2))) \/ 1 \\ Charles R Greathouse IV, Jun 16 2021
A058381
Number of series-parallel networks with n labeled edges, multiple edges not allowed.
Original entry on oeis.org
0, 1, 1, 4, 20, 156, 1472, 17396, 239612, 3827816, 69071272, 1394315088, 31081310944, 758901184432, 20135117147056, 576927779925568, 17752780676186432, 583910574851160000, 20443098012485430272, 759064322969950283072, 29793617955495321025472
Offset: 0
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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 *)
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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 */
A058388
Total number of interior nodes in all essentially parallel series-parallel networks with n labeled edges, multiple edges not allowed.
Original entry on oeis.org
0, 0, 0, 3, 14, 195, 2059, 31150, 489012, 9073638, 183490118, 4135560660, 101421574440, 2706766547628, 77860733488732, 2405136817507216, 79353915366944784, 2786110796782734528, 103703080088989729280, 4079350129335095498048
Offset: 0
- J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_Q(n)*Q_pi).
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max = 19; q = CoefficientList[ InverseSeries[ Series[-1 + E^(1 + 2*a - E^a), {a, 0, max}], x], x]*Table[x^k, {k, 0, max}] // Total; r = q - Log[1 + x]; v = q + r; ev = (v*q - r)/(1 - v); eq = (ev + v)/(1 + v) - q; CoefficientList[ Series[eq, {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Feb 01 2013 *)
A058406
Total number of interior nodes in all series-parallel networks with n labeled edges, multiple edges not allowed.
Original entry on oeis.org
0, 0, 1, 2, 27, 199, 2645, 34236, 560742, 9958754, 201928954, 4480386932, 109410252512, 2897637649204, 82974026800132, 2550731142019568, 83843131420325008, 2933465366569951168, 108862752438362487648, 4270766898251635808800
Offset: 0
- J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_R(n)*Q_pi).
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max = 19; q = CoefficientList[ InverseSeries[ Series[-1 + E^(1 + 2*a - E^a), {a, 0, max}], x], x]*Table[x^k, {k, 0, max}] // Total; r = q - Log[1 + x]; v = q + r; ev = (v*q - r)/(1 - v); er = ev/(1 + v); CoefficientList[ Series[er, {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Feb 01 2013 *)
A058475
Total number of interior nodes in all series-parallel networks with n labeled edges, multiple edges not allowed.
Original entry on oeis.org
0, 0, 1, 5, 41, 394, 4704, 65386, 1049754, 19032392, 385419072, 8615947592, 210831826952, 5604404196832, 160834760288864, 4955867959526784, 163197046787269792, 5719576163352685696, 212565832527352216928, 8350117027586731306848
Offset: 0
- J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_V(n)*Q_pi).
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max = 19; q = CoefficientList[ InverseSeries[ Series[-1 + E^(1 + 2*a - E^a), {a, 0, max}], x], x]*Table[x^k, {k, 0, max}] // Total; r = q - Log[1 + x]; v = q + r; ev = (v*q - r)/(1 - v); CoefficientList[ Series[ev, {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Feb 01 2013 *)
A339299
Number of essentially series oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.
Original entry on oeis.org
0, 2, 6, 72, 840, 14040, 276360, 6494880, 175452480, 5375311200, 183962227680, 6958070380800, 288200792880000, 12974113884251520, 630742839699772800, 32933429270386444800, 1838083950894102912000, 109201772719684867622400, 6880730833827011402841600
Offset: 1
A058349 is the case with multiple unit elements in parallel allowed.
A058380 is the case that order is not significant in series configurations.
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seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = (1 + Z)*exp(p^2/(1+p)) - 1); Vec(serlaplace(p-p/(1+p)), -n)}
Showing 1-6 of 6 results.
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