cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

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

Views

Author

N. J. A. Sloane, Dec 19 2000

Keywords

Links

Crossrefs

Equals A058379 + A058380.
Cf. A006351.

Programs

  • 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 *)
  • 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 */

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

A058380 Essentially series series-parallel networks with n labeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 0, 1, 1, 13, 66, 796, 8338, 122326, 1893748, 34717076, 695343144, 15560613872, 379211091416, 10070672083928, 288420300817184, 8877044175277216, 291944826030636000, 10221726849956763136, 379528960298122277536, 14896869800297864928736
Offset: 0

Views

Author

N. J. A. Sloane, Dec 19 2000

Keywords

Links

Crossrefs

Cf. A058379, A058381.

Programs

  • Mathematica
    CoefficientList[Series[-1/2 - Log[1+x]/2 - LambertW[-E^(-1/2)*Sqrt[1+x]/2], {x, 0, 15}], x]* Range[0, 15]! (* Vaclav Kotesovec, Mar 11 2014 *)

Formula

E.g.f. satisfies A(x) = A058379(x) - log(1+x).
E.g.f.: -1/2 - log(1+x)/2 - LambertW(-exp(-1/2)*sqrt(1+x)/2). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ n^(n-1) / (2*sqrt(2)*(4-exp(1))^(n-1/2)). - Vaclav Kotesovec, Mar 11 2014

A058385 Number of essentially parallel series-parallel networks with n unlabeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 1, 0, 1, 2, 4, 9, 20, 47, 112, 274, 678, 1709, 4346, 11176, 28966, 75656, 198814, 525496, 1395758, 3723986, 9975314, 26817655, 72332320, 195679137, 530814386, 1443556739, 3934880554, 10748839215, 29420919456, 80678144437, 221618678694
Offset: 0

Views

Author

N. J. A. Sloane, Dec 20 2000

Keywords

Links

Crossrefs

Cf. A058379, A058386, A058387.

Programs

  • Maple
    Q := x; q[1] := 1; for d from 1 to 40 do q[d+1] := c; Q := Q+c*x^(d+1); t0 := mul((1-x^j)^(-q[j]),j=1..d+1); t01 := series(t0,x,d+2); t05 := series(2*Q +1-x+x^2 -t01, x, d+2); t1 := coeff(t05,x,d+1); t2 := solve(t1,c); q[d+1] := t2; Q := subs(c=t2,Q); Q := series(Q,x,d+2); od: A058385 := n->coeff(Q,x,n);
  • Mathematica
    max = 31; f[x_] := Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 0; a[3] = 1; se = Series[ 1 - x + x^2 + 2*f[x] - Product[(1 - x^j)^(-a[j]), {j, 1, max}], {x, 0, max}]; sol = Solve[ Thread[ CoefficientList[ se, x] == 0]]; A058385 = Table[a[n], {n, 0, max}] /. First[sol] (* Jean-François Alcover, Dec 27 2011, after g.f. *)
terms = 32; A[] = 0; Do[A[x] = (1/2)*(-1 + x - x^2 + Product[(1 - x^j)^(-Ceiling[Coefficient[A[x], x, j]]), {j, 1,  terms}]) + O[x]^ terms // Normal, 4*terms]; CoefficientList[A[x] + O[x]^terms, x] (* Jean-François Alcover, Jan 10 2018 *)

Formula

G.f. satisfies 1 - x + x^2 + 2*A(x) = Product_{j>=1} (1-x^j)^(-a(j)).

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

Views

Author

N. J. A. Sloane, Dec 20 2000

Keywords

References

  • J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_Q(n)*Q_pi).

Links

Programs

  • Mathematica
    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 *)

Formula

Let Q, R = Q-log(1+x), V=Q+R be the e.g.f.'s for A058379, A058380, A058381 resp. E.g.f.'s for A058475, A058406, A058388 are E_V = (V*Q-R)/(1-V), E_R = E_V/(1+V), E_Q = (E_V+V)/(1+V)-Q.

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

Views

Author

N. J. A. Sloane, Dec 20 2000

Keywords

References

  • J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_R(n)*Q_pi).

Links

Programs

  • Mathematica
    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 *)

Formula

Let Q, R = Q-log(1+x), V=Q+R be the e.g.f.'s for A058379, A058380, A058381 resp. E.g.f.'s for A058475, A058406, A058388 are E_V = (V*Q-R)/(1-V), E_R = E_V/(1+V), E_Q = (E_V+V)/(1+V)-Q.

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

Views

Author

N. J. A. Sloane, Dec 20 2000

Keywords

References

  • J. W. Moon, Some enumerative results on series-parallel networks, Annals Discrete Math., 33 (1987), 199-226 (the sequence I_V(n)*Q_pi).

Links

Programs

  • Mathematica
    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 *)

Formula

Let Q, R = Q-log(1+x), V=Q+R be the e.g.f.'s for A058379, A058380, A058381 resp. E.g.f.'s for A058475, A058406, A058388 are E_V = (V*Q-R)/(1-V), E_R = E_V/(1+V), E_Q = (E_V+V)/(1+V)-Q.

A058386 Essentially series series-parallel networks with n unlabeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 9, 20, 47, 112, 274, 678, 1709, 4346, 11176, 28966, 75656, 198814, 525496, 1395758, 3723986, 9975314, 26817655, 72332320, 195679137, 530814386, 1443556739, 3934880554, 10748839215, 29420919456, 80678144437, 221618678694
Offset: 0

Views

Author

N. J. A. Sloane, Dec 20 2000

Keywords

Links

Crossrefs

Cf. A058379, A058385, A058387.

Programs

  • Mathematica
    (* f = g.f. of A058385 *) max = 31; f[x_] := Sum[b[n]*x^n, {n, 0, max}]; b[0] = 0; b[1] = 1; b[2] = 0; b[3] = 1; coef = CoefficientList[ Series[1 - x + x^2 + 2*f[x] - Product[(1 - x^j)^(-b[j]), {j, 1, max}], {x, 0, max}], x][[ 5 ;; All]]; g[x_] := Sum[a[n]*x^n, {n, 0, max}]; a[0] = a[1] = 0; a[2] = a[3] = 1; coeg = CoefficientList[ Series[g[x] - f[x] + x - x^2, {x, 0, max}], x][[ 5 ;; All]]; solf = SolveAlways[ Thread[coef == 0], x] ; solg = SolveAlways[ Thread[coeg == 0] /. solf[[1]], x]; Table[a[n], {n, 0, max}] /. solg[[1]] (* Jean-François Alcover, Jul 18 2012 *)
terms = 32; (* f = g.f. of A058385 *) f[] = 0; Do[f[x] = (1/2)*(-1 + x - x^2 + Product[(1 - x^j)^(-Ceiling[Coefficient[f[x], x, j]]), {j, 1,  terms}]) + O[x]^ terms // Normal, 4*terms]; A[x_] = f[x] - x + x^2 + O[x]^terms; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 10 2018 *)

Formula

G.f. satisfies A(x) = A058385(x) - x + x^2.

A339300 Number of essentially parallel oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 0, 6, 36, 540, 8400, 169680, 3966480, 107518320, 3295283040, 112888369440, 4272403544640, 177061349424960, 7974538914101760, 387840385867334400, 20257533315635616000, 1130954856127948051200, 67208532822729871372800, 4235759061057115720128000
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2020

Keywords

Comments

See A339301 for additional details.

Crossrefs

A048174 is the case with multiple edges in parallel allowed.
A058379 is the case that order is not significant in series configurations.
Cf. A339289 (unlabeled), A339299, A339301.

Programs

  • PARI
    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(1-1/(1+p)))}

Formula

E.g.f. (1 + x)*exp(S(x)) - S(x) - 1 where S(x) is the e.g.f. of A339299.
E.g.f.: B(x)/(1 + B(x)) where B(x) is the e.g.f. of A339301.
E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048174.
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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