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 11-19 of 19 results.

A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes.

Original entry on oeis.org

10, 840, 257040, 137260200, 118273755600, 154712104747200, 292311804557572800, 766931112143320924800, 2706462791802644002128000, 12512595130808078973370704000, 74130965352250071944327288640000, 552334353713465817349513210512960000, 5092566798555894395129552704613028960000
Offset: 3

Views

Author

N. J. A. Sloane

Keywords

Comments

R. C. Read incorrectly has a(7) = 118257539400 and a(8) = 154678050727200 which he calculated by hand. - Sean A. Irvine, Jun 27 2017

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A001501, A002829, A004109, A246599.

Programs

  • Mathematica
    (* b stands for A001501 *) b[n_] := n!^2 Sum[2^(2k-n) 3^(k-n) (3(n-k))!  HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k)!^2), {k, 0, n}]/6^n;
(* c stands for A246599 *) c[n_] := c[n] = Binomial[2n-1, n] b[n] - Sum[ Binomial[2n-1, 2k] Binomial[2k, k] b[k] c[n-k], {k, 1, n-1}];
a[n_] := a[n] = c[n] + Sum[Binomial[2n-1, 2k-1] c[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)
  • PARI
    \\ here b(n) is A001501
b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))}
seq(n)={my(v=vector(n,n,b(n)*binomial(2*n-1,n)), u=vector(n), s=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, 2*binomial(2*n-1,2*k)*v[k]*u[n-k]); s[n]=u[n] + sum(k=3, n-3, binomial(2*n-1,2*k-1)*u[k]*s[n-k])); s[3..n]} \\ Andrew Howroyd, May 22 2018

Formula

a(n) = A246599(n) + Sum_{k=1..n-1} binomial(2*n-1,2*k-1)*A246599(k)*a(n-k). - Andrew Howroyd, May 22 2018
a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n+2)). - Vaclav Kotesovec, Feb 17 2024

Extensions

a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017
Terms a(13) and beyond from Andrew Howroyd, May 22 2018

A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.

Original entry on oeis.org

10, 840, 257040, 137214000, 118248530400, 154686980448000, 292276881344448000, 766864651478365440000, 2706292794907249067520000, 12512021073989410699165440000, 74128448237031250090060032000000, 552320243355746711191770103680000000, 5092467146398443040845772685937408000000
Offset: 3

Views

Author

N. J. A. Sloane, Sep 08 2014

Keywords

Comments

R. C. Read incorrectly has a(7) = 118237555800 and a(8) = 154652926428000 which he calculated by hand.

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Links

Crossrefs

Cf. A001501, A002829, A004109, A006714.

Programs

  • Mathematica
    b[n_] := n!^2*Sum[2^(2k-n) 3^(k-n)(3(n-k))!*HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k )!^2), {k, 0, n}]/6^n;
a[n_] := a[n] = Binomial[2n-1, n] b[n] - Sum[Binomial[2n-1, 2k] Binomial[2 k, k] b[k] a[n-k], {k, 1, n-1}];
Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *)
  • PARI
    \\ here b(n) is A001501
b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))}
seq(n)={my(v=vector(n, n, b(n)*binomial(2*n, n)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, binomial(2*n-1,2*k)*v[k]*u[n-k])); u[3..n]/2} \\ Andrew Howroyd, May 22 2018

Formula

a(n) = binomial(2*n-1, n)*A001501(n) - Sum_{k=1..n-1} binomial(2*n-1, 2*k) * binomial(2*k, k) * A001501(k) * a(n-k). - Andrew Howroyd, May 22 2018
a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n + 2)). - Vaclav Kotesovec, Feb 17 2024

Extensions

a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017
Terms a(13) and beyond from Andrew Howroyd, May 22 2018

A246970 Number of connected bicubical multigraphs on 2n labeled nodes of two colors.

Original entry on oeis.org

1, 2, 31, 1272, 105720
Offset: 1

Views

Author

N. J. A. Sloane, Sep 09 2014

Keywords

Comments

Read's 1971 letter includes additional terms for both this sequence and A001500. But the terms a(6) and a(7) given for A001500 are wrong, so presumably those for this sequence are also wrong. See A246968 and A246969 for the sequences as given in Read's letter and (presumably) also in his dissertation.

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Links

Crossrefs

Cf. A001500, A246968, A246969.

A246968 Erroneous version of A001500.

Original entry on oeis.org

1, 4, 55, 2008, 153040, 20987840, 4672874360
Offset: 1

Views

Author

N. J. A. Sloane, Sep 09 2014

Keywords

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Links

Crossrefs

Cf. A001500.

A246969 Erroneous version of A246970.

Original entry on oeis.org

1, 2, 31, 1272, 105720, 15492600, 3621844800
Offset: 1

Views

Author

N. J. A. Sloane, Sep 09 2014

Keywords

Comments

This is the "connected" version of the problem studied by Read in A001500. Since the terms a(6) and a(7) for A001500 given in his letter (and presumably also in his dissertation) are known to be wrong, it is extremely likely that the values of a(6) and a(7) shown here are also wrong.

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 195

Links

A248361 Erroneous version of A006712.

Original entry on oeis.org

6, 480, 197820, 154103040, 215643443400
Offset: 2

Views

Author

Sean A. Irvine, Oct 05 2014

Keywords

Comments

Included in accordance with the OEIS policy of listing published but incorrect sequences, to serve as pointers to the correct entries.

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

A248362 Erroneous version of A006713.

Original entry on oeis.org

6, 480, 196560, 153498240, 214951968000
Offset: 2

Views

Author

Sean A. Irvine, Oct 05 2014

Keywords

Comments

Included in accordance with the OEIS policy of listing published but incorrect sequences, to serve as pointers to the correct entries.

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

A004577 Erroneous version of A000940.

Original entry on oeis.org

1, 2, 4, 7, 39, 202, 1219, 9468, 83435, 80017
Offset: 3

Views

Author

N. J. A. Sloane, Sep 09 2014

Keywords

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

Links

A246602 Number of connected bicubical graphs on 2n labeled nodes of two colors.

Original entry on oeis.org

1, 24, 2040, 297000, 68922000, 24038380800
Offset: 3

Views

Author

N. J. A. Sloane, Sep 08 2014

Keywords

Comments

Should a(7) be 68909400? - Andrew Howroyd, May 23 2018

References

  • R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

Links

Crossrefs

Cf. A001501.
Previous Showing 11-19 of 19 results.

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

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