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 1-7 of 7 results.

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

Views

Author

N. J. A. Sloane, Dec 19 2000

Keywords

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

Crossrefs

Cf. A058380, A058381. See A000669 for unlabeled case when parallel edges are allowed.

Programs

  • 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
  • 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 */
  • 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

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

A058387 Number of series-parallel networks with n unlabeled edges, multiple edges not allowed.

Original entry on oeis.org

0, 1, 1, 2, 4, 8, 18, 40, 94, 224, 548, 1356, 3418, 8692, 22352, 57932, 151312, 397628, 1050992, 2791516, 7447972, 19950628, 53635310, 144664640, 391358274, 1061628772, 2887113478, 7869761108, 21497678430, 58841838912, 161356288874
Offset: 0

Views

Author

N. J. A. Sloane, Dec 20 2000

Keywords

Comments

This is a series-parallel network: o-o; all other series-parallel networks are obtained by connecting two series-parallel networks in series or in parallel. See A000084 for examples.
Order is not considered significant in series configurations. - Andrew Howroyd, Dec 22 2020

Examples

			From _Andrew Howroyd_, Dec 22 2020: (Start)
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element (an edge) is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 8: (ooooo), (oo(o|oo)), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)),  (oo|ooo), (o|oo|oo).
(End)

Links

Crossrefs

A000084 is the case that multiple edges are allowed.
A058381 is the case that edges are labeled.
A339290 is the case that order is significant in series configurations.
Cf. A058385, A058386, A000311, A000669, A006351.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(s=p=vector(n)); p[1]=1; for(n=2, n, s[n]=EulerT(p[1..n])[n]; p[n]=vecsum(EulerT(s[1..n])[n-1..n])-s[n]); concat([0], p+s)} \\ Andrew Howroyd, Dec 22 2020

Formula

a(n) = A058385(n) + A058386(n).

A339301 Number of oriented series-parallel networks with n labeled elements and without multiple unit elements in parallel.

Original entry on oeis.org

1, 2, 12, 108, 1380, 22440, 446040, 10461360, 282970800, 8670594240, 296850597120, 11230473925440, 465262142304960, 20948652798353280, 1018583225567107200, 53190962586022060800, 2969038807022050963200, 176410305542414738995200, 11116489894884127122969600
Offset: 1

Views

Author

Andrew Howroyd, Dec 22 2020

Keywords

Comments

A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. a(n) is the total number of series and parallel configurations with n unit elements labeled 1..n.

Examples

			a(3) = 12 because there are 2 unlabeled structures each of which can be labeled in 6 ways. The unlabeled structures are (ooo) and (o|oo).

Crossrefs

A048172 is the case with multiple unit elements in parallel allowed.
A058381 is the case that order is not significant in series configurations.
Main diagonal of A339297.
Cf. A339290 (unlabeled), A339299, A339300.

Programs

  • PARI
    \\ Note giving Z=exp(x)-1 gives A048172.
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))}
  • PARI
    seq(n)={my(A=O(x*x^n)); Vec(serlaplace(subst(serreverse(log(1+x+A) - x^2/(1+x)), x, log(1+x+A))))}

Formula

a(n) = A339299(n) + A339300(n).
E.g.f.: A(x) satisfies A(x) = (1 + x)*exp(A(x)^2/(1+A(x))) - 1.
E.g.f.: P(x)/(1 - P(x)) where P(x) is the e.g.f. of A339300.
E.g.f.: B(log(1+x)) where B(x) is the e.g.f. of A048172.

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.
Showing 1-7 of 7 results.

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

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