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 1-3 of 3 results.

A058371 The sequence S defined in A058540.

Original entry on oeis.org

0, 0, 1, 3, 12, 48, 217, 1005, 4878, 24218, 123021, 634995, 3323963, 17596131, 94053675, 506865965, 2751093957, 15025119759, 82511460766, 455331632088, 2523688452540, 14042621802676, 78415665739800, 439298161502112, 2468288819015277
Offset: 0

Views

Author

N. J. A. Sloane, Dec 26 2000

Keywords

Crossrefs

Cf. A058575.

Programs

  • Maple
    spec := [ S, {N=Union(Z,S,P,Q), S=Set(Union(Z,P,Q),card>=2), P=Set(Union(Z,S,Q),card>=2), Q=Set(Union(Z,S,P),card>=2)} ]; [seq(combstruct[count](spec,size=n), n=0..40)]; # N=A058540, S=A058371
spec:=[S,{S=Set(Union(Z,S,S),card>=2)}];[seq(combstruct[count](spec,size=n),n=0..25)]; # Vladeta Jovovic, Jun 25 2007

A058562 Another 3-way generalization of series-parallel networks with n labeled edges.

Original entry on oeis.org

0, 1, 3, 21, 243, 3933, 81819, 2080053, 62490339, 2166106509, 85092601707, 3735939709989, 181287330220467, 9634718677393917, 556569415611455931, 34723276781195740437, 2326773811332029313411, 166666995789875216053101, 12708546598923724476443403
Offset: 0

Views

Author

N. J. A. Sloane, Dec 26 2000

Keywords

Links

Crossrefs

Cf. A058540, A058371, A058575, A201465.

Programs

  • Maple
    spec := [ N, {N=Union(Z,S,P,Q), S=Set(Union(Z,P,Q),card>=2), P=Set(Union(Z,S,Q),card>=2), Q=Set(Union(Z,S,P),card>=2)}, labeled ]; [seq(combstruct[count](spec,size=n), n=0..40)]; # N=A058562, S=A058575
# Alternatively:
A058562_list := proc(len) local A, n; A[0] := 0; A[1] := 1; for n from 2 to len do
A[n] := A[n-1] + add(binomial(n,j)*A[j]*A[n-j], j=1..n-1) od:
convert(A,list) end: A058562_list(18); # Peter Luschny, May 24 2017
  • Mathematica
    a[n_] := Sum[(n+k-1)!*Sum[1/(k-j)!*Sum[(3^(j-l)*(2)^l*(-1)^(l+j)* StirlingS1[n-l+j-1, j-l])/(l!*(n-l+j-1)!), {l, 0, j}], {j, 0, k}], {k, 0, n-1}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
  • Maxima
    a(n):=sum((n+k-1)!*sum(1/(k-j)!*sum((3^(j-l)*(2)^l*(-1)^(l+j)*stirling1(n-l+j-1,j-l))/(l!*(n-l+j-1)!),l,0,j),j,0,k),k,0,n-1); /* Vladimir Kruchinin, Sep 26 2012 */
  • PARI
    {a(n)=if(n<1,0,n!*polcoeff(serreverse(3*log(1+x+x*O(x^n))-2*x),n))} \\ Paul D. Hanna, Aug 03 2008

Formula

E.g.f.: -3/2*LambertW(-2/3*exp(-2/3+1/3*x))-1. - Vladeta Jovovic, Jun 25 2007
E.g.f.: A(x) = Series_Reversion[ 3*log(1+x) - 2*x ]. [Paul D. Hanna, Aug 03 2008]
Let f(x) = (1+x)/(1-2*x). Let D be the operator g(x) -> d/dx(f(x)*g(x)). Then for n>=1, a(n) = D^(n-1)(1) evaluated at x = 0. - Peter Bala, Sep 05 2011
log(1 + A(x)) = x + 2*x^2/2! + 14*x^3/3! + 162*x^4/4! + ... is the e.g.f. for A201465. - Peter Bala, Jul 12 2012
a(n) = sum(k=0..n-1, (n+k-1)!*sum(j=0..k, 1/(k-j)!*sum(l=0..j, (3^(j-l)*(2)^l*(-1)^(l+j)*stirling1(n-l+j-1,j-l))/(l!*(n-l+j-1)!)))). [Vladimir Kruchinin, Sep 26 2012]
G.f.: x/Q(0), where Q(k)= 1 - (k+1)*x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
a(n) ~ sqrt(3) * n^(n-1) / (2*exp(n) * (log(27/8)-1)^(n-1/2)). - Vaclav Kotesovec, Oct 05 2013
a(n) = a(n-1) + Sum_{j=1..n-1} binomial(n,j)*a(j)*a(n-j) for n>1. - Peter Luschny, May 24 2017

A058534 A 3-way generalization of series-parallel networks with n unlabeled edges.

Original entry on oeis.org

0, 1, 3, 6, 15, 36, 99, 270, 783, 2298, 6936, 21204, 65895, 206862, 656253, 2098602, 6761028, 21917364, 71450229, 234070806, 770216253, 2544458592, 8435990916, 28060099692, 93612265143, 313153860210, 1050194570445, 3530080085868
Offset: 0

Views

Author

N. J. A. Sloane, Dec 24 2000

Keywords

Comments

Compare the combstruct construction here with that for A000084.

Crossrefs

Cf. A000084, A058540, A000669, A058561.

Programs

  • Maple
    spec := [ N, {N=Union(Z,S,P,Q), S=Set(Union(Z,P),card>=2), P=Set(Union(Z,Q),card>=2), Q=Set(Union(Z,S),card>=2)} ]; [seq(combstruct[count](spec,size=n), n=0..40)]; # N = A058534, S=A000669
Showing 1-3 of 3 results.

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

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