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.

Previous Showing 81-83 of 83 results.

A283825 Number of Hamiltonian regular graphs on n nodes.

Original entry on oeis.org

1, 0, 1, 2, 2, 5, 4, 17, 22, 165, 538, 18972, 389426, 50314715
Offset: 1

Views

Author

N. J. A. Sloane, Mar 19 2017

Keywords

Comments

By convention, the singleton graph is generally considered to be both regular (cf. A005176) and Hamiltonian (cf. A003216). - Eric W. Weisstein, Oct 30 2017

Links

Crossrefs

Cf. A005176, A005177.

Extensions

a(11)-a(14) added using tinygraph by Falk Hüffner, Mar 31 2017
a(1) changed from 0 to 1 by Eric W. Weisstein, Oct 30 2017

A333865 Number of simple graphs on n nodes with vertex count > edge count + 1.

Original entry on oeis.org

0, 1, 2, 4, 8, 18, 40, 100, 256, 705, 2057, 6370, 20803, 71725, 259678, 985244, 3905022, 16124936, 69188809, 307765510, 1416146859, 6727549181, 32938379216, 165942445714, 859020421012, 4563322971706, 24847598243116, 138533012486423, 790075521708603, 4605183081182354
Offset: 1

Views

Author

Eric W. Weisstein, Apr 08 2020

Keywords

Comments

These graphs correspond to "trivially ungraceful" graphs that do not have enough integers less than or equal to the edge count to cover all the vertices.

Links

Crossrefs

Cf. A008406.
Cf. A308556 (number of simple ungraceful graphs on n nodes).

Programs

  • Mathematica
    Get["Combinatorica`"] // Quiet;
Table[Total[Take[CoefficientList[GraphPolynomial[n, x], x], n - 1]], {n, 20}]
  • PARI
    permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
a(n)={my(s=0); if(n>1, forpart(p=n, s+=permcount(p)*polcoef(edges(p, i->1 + x^i + O(x^(n-1)))/(1-x), n-2) )); s/n!} \\ Andrew Howroyd, Apr 08 2020

Formula

a(n) <= A308556(n).
a(n) = Sum_{k=0..n-2} A008406(n, k). - Andrew Howroyd, Apr 08 2020

Extensions

Terms a(11) and beyond from Andrew Howroyd, Apr 08 2020

A354607 Triangular array read by rows: T(n,k) is the number of labeled tournaments on [n] that have exactly k irreducible (strongly connected) components, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 2, 0, 6, 0, 24, 16, 0, 24, 0, 544, 240, 120, 0, 120, 0, 22320, 6608, 2160, 960, 0, 720, 0, 1677488, 315840, 70224, 20160, 8400, 0, 5040, 0, 236522496, 27001984, 3830400, 758016, 201600, 80640, 0, 40320, 0, 64026088576, 4268194560, 366729600, 46448640, 8628480, 2177280, 846720, 0, 362880
Offset: 0

Views

Author

Geoffrey Critzer, Jul 08 2022

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     0,    2;
  0,     2,    0,    6;
  0,    24,   16,    0,  24;
  0,   544,  240,  120,   0, 120;
  0, 22320, 6608, 2160, 960,   0, 720;
  ...

Links

Crossrefs

Cf. A006125 (row sums), A054946 (column k=1), A000142 (main diagonal).

Programs

  • Mathematica
    nn = 10; G[x_] := Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; Table[
Take[(Range[0, nn]! CoefficientList[Series[1/(1 - y (1 - 1/ G[x])), {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}]

Formula

E.g.f.: 1/(1-y*(1-1/A(x))) where A(x) is the e.g.f. for A006125.
Previous Showing 81-83 of 83 results.

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

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