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 1-2 of 2 results.

A095983 Number of 2-edge-connected labeled graphs on n nodes.

Original entry on oeis.org

0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016
Offset: 0

Views

Author

Yifei Chen (yifei(AT)mit.edu), Jul 17 2004

Keywords

Comments

From Falk Hüffner, Jun 28 2018: (Start)
Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are k-edge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
Labeled version of A007146. (End)
The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. This sequence counts graphs with spanning edge-connectivity >= 2, which, for n > 1, are connected graphs with no bridges. - Gus Wiseman, Sep 20 2019

Links

Crossrefs

Cf. A053549, A198046.
The unlabeled version is A007146.
Row sums of A327069 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with spanning edge-connectivity 2 are A327146.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Graphs without endpoints are A059167.
Graphs with spanning edge-connectivity 1 are A327071.
Cf. A001187, A002218, A052446, A059166, A100743, A322395, A327079, A327144.

Programs

  • Mathematica
    seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k-1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
seq[16] (* Jean-François Alcover, Aug 13 2019, after Andrew Howroyd *)
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]>=2&]],{n,0,5}] (* Gus Wiseman, Sep 20 2019 *)
  • PARI
    \\ here p is initially A053549, q is A198046 as e.g.f.s.
seq(n)={my(v=vector(n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
my(q=x*exp(p)); p-=q;
for(k=3, n, my(c=polcoeff(p,k)); v[k]=c*(k-1)!; p-=c*q^k);
concat([0],v)} \\ Andrew Howroyd, Jun 18 2018
  • PARI
    seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x-1), -(n+1))} \\ Andrew Howroyd, Dec 28 2020

Formula

a(n) = A001187(n) - A327071(n). - Gus Wiseman, Sep 20 2019

Extensions

Name corrected and more terms from Pavel Irzhavski, Nov 01 2014
Offset corrected by Falk Hüffner, Jun 17 2018
a(12)-a(16) from Andrew Howroyd, Jun 18 2018

A274804 The exponential transform of sigma(n).

Original entry on oeis.org

1, 1, 4, 14, 69, 367, 2284, 15430, 115146, 924555, 7991892, 73547322, 718621516, 7410375897, 80405501540, 914492881330, 10873902417225, 134808633318271, 1738734267608613, 23282225008741565, 323082222240744379, 4638440974576329923, 68794595993688306903
Offset: 0

Views

Author

Johannes W. Meijer, Jul 27 2016

Keywords

Comments

The exponential transform [EXP] transforms an input sequence b(n) into the output sequence a(n). The EXP transform is the inverse of the logarithmic transform [LOG], see the Weisstein link and the Sloane and Plouffe reference. This relation goes by the name of Riddell's formula. For information about the logarithmic transform see A274805. The EXP transform is related to the multinomial transform, see A274760 and the second formula.
The definition of the EXP transform, see the second formula, shows that n >= 1. To preserve the identity LOG[EXP[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the exponential transform, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A178867 appear.
We observe that a(0) = 1 and provides no information about any value of b(n), this notwithstanding it is customary to start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the exponential transform of a sequence. The first program uses a formula found by Alois P. Heinz, see A007446 and the first formula. The second program uses the definition of the exponential transform, see the Weisstein link and the second formula. The third program uses information about the inverse of the exponential transform, see A274805.
Some EXP transform pairs are, n >= 1: A000435(n) and A065440(n-1); 1/A000027(n) and A177208(n-1)/A177209(n-1); A000670(n) and A075729(n-1); A000670(n-1) and A014304(n-1); A000045(n) and A256180(n-1); A000290(n) and A033462(n-1); A006125(n) and A197505(n-1); A053549(n) and A198046(n-1); A000311(n) and A006351(n); A030019(n) and A134954(n-1); A038048(n) and A053529(n-1); A193356(n) and A003727(n-1).

Examples

			Some a(n) formulas, see A178867:
a(0) = 1
a(1) = x(1)
a(2) = x(1)^2 + x(2)
a(3) = x(1)^3 + 3*x(1)*x(2) + x(3)
a(4) = x(1)^4 + 6*x(1)^2*x(2) + 4*x(1)*x(3) + 3*x(2)^2 + x(4)
a(5) = x(1)^5 + 10*x(1)^3*x(2) + 10*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 5*x(1)*x(4) + 10*x(2)*x(3) + x(5)

References

  • Frank Harary and Edgar M. Palmer, Graphical Enumeration, 1973.
  • Robert James Riddell, Contributions to the theory of condensation, Dissertation, University of Michigan, Ann Arbor, 1951.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

Links

Crossrefs

Cf. A274805, A274760, A178867, A000203, A007446.
Cf. A177208, A177209, A006351, A197505, A144180, A256180, A033462, A198046, A134954, A145460, A188489, A005432, A029725, A124213, A002801.
Cf. A036040, another version.

Programs

  • Maple
    nmax:=21: with(numtheory): b := proc(n): sigma(n) end: a:= proc(n) option remember; if n=0 then 1 else add(binomial(n-1, j-1) * b(j) *a(n-j), j=1..n) fi: end: seq(a(n), n=0..nmax); # End first EXP program.
nmax:= 21: with(numtheory): b := proc(n): sigma(n) end: t1 := exp(add(b(n)*x^n/n!, n=1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n=0..nmax); # End second EXP program.
nmax:=21: with(numtheory): b := proc(n): sigma(n) end: f := series(log(1+add(q(n)*x^n/n!, n=1..nmax+1)), x, nmax+1): d := proc(n): n!*coeff(f, x, n) end: a(0):=1: q(0):=1: a(1):=b(1): q(1):=b(1): for n from 2 to nmax+1 do q(n) := solve(d(n)-b(n), q(n)): a(n):=q(n): od: seq(a(n), n=0..nmax); # End third EXP program.
  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[Binomial[n-1, j-1]*DivisorSigma[1, j]*a[n-j], {j, 1, n}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 22 2017 *)
nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 08 2021 *)

Formula

a(n) = Sum_{j=1..n} (binomial(n-1,j-1) * b(j) * a(n-j)), n >= 1 and a(0) = 1, with b(n) = A000203(n) = sigma(n).
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n!) with b(n) = sigma(n) = A000203(n).
Showing 1-2 of 2 results.

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