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 1-8 of 8 results.

A333330 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n unlabeled nodes, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 3, 2, 1, 1, 1, 0, 1, 0, 4, 0, 4, 0, 1, 1, 0, 1, 1, 5, 7, 9, 4, 1, 1, 1, 0, 1, 0, 7, 0, 24, 0, 7, 0, 1, 1, 0, 1, 1, 8, 16, 54, 60, 32, 8, 1, 1, 1, 0, 1, 0, 10, 0, 128, 0, 240, 0, 12, 0, 1, 1, 0, 1, 1, 12, 37, 271, 955, 1753, 930, 135, 14, 1, 1
Offset: 0

Views

Author

Andrew Howroyd, Mar 15 2020

Keywords

Comments

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A333351. Burnside's lemma can be used to extend this method to the unlabeled case.

Examples

			Array begins:
=================================================
n\k | 0 1 2  3   4    5      6     7        8
----+--------------------------------------------
  0 | 1 1 1  1   1    1      1     1        1 ...
  1 | 1 0 0  0   0    0      0     0        0 ...
  2 | 1 1 1  1   1    1      1     1        1 ...
  3 | 1 0 1  0   1    0      1     0        1 ...
  4 | 1 1 2  3   4    5      7     8       10 ...
  5 | 1 0 2  0   7    0     16     0       37 ...
  6 | 1 1 4  9  24   54    128   271      582 ...
  7 | 1 0 4  0  60    0    955     0    12511 ...
  8 | 1 1 7 32 240 1753  13467 90913   543779 ...
  9 | 1 0 8  0 930    0 253373     0 35255015 ...
  ...

Links

Crossrefs

Columns k=0..8 are (with interspersed 0's for odd k): A000012, A000012, A002865, A129416, A129418, A129420, A129422, A129424, A129426.
Row n=4 is A001399.
Cf. A051031 (simple graphs), A167625 (with loops), A192517 (not necessarily regular), A328682 (connected), A333351 (labeled nodes).

A129416 Number of isomorphism classes of 3-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 3, 9, 32, 135, 709, 4637, 38374, 391473, 4764778, 66913591, 1056886475, 18446472265, 351482430368, 7247888726269, 160671989129665, 3808499268504548, 96094161981827499, 2570930535917564366, 72688753062897675445
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Links

Crossrefs

Column k=3 of A333330.
Cf. A000421 (connected, inv. Eul. trans.), A129427, A129418, A129420, A129422, A129424, A129426.

Formula

Euler transform of A000421.

Extensions

a(13)-a(20) from Andrew Howroyd, Mar 19 2020

A129421 Number of isomorphism classes of connected 6-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 6, 15, 120, 933, 13303, 252207, 6450828, 205475039, 7936493756, 363639228194, 19476976825809, 1205115679461426, 85288127619421544, 6845235025444882069, 618411485467843477405, 62471139399366989007575, 7014991719815977343879171
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Links

Crossrefs

Column k=6 of A328682.
Cf. A129422, A129432, A000421, A129417, A129419, A129423, A129425, A328682.

Programs

Formula

Inverse Euler transform of A129422. - Andrew Howroyd, Mar 17 2020

Extensions

a(1)=0 prepended and a(14)-a(20) from Andrew Howroyd, Mar 17 2020

A129426 Number of isomorphism classes of 8-regular loopless multigraphs of order n.

Original entry on oeis.org

0, 1, 1, 10, 37, 582, 12511, 543779, 35255015, 3230979297, 397550237967, 63834143947661, 13080849749829233, 3358751856150607392, 1063851391062768324862, 410060430118305494628648, 190065946515113295597969794, 104826174445642584491349328181
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=8 of A333330.
Cf. A129425, A129437, A129416, A129418, A129420, A129422, A129424.

Formula

Euler transform of A129425. - Andrew Howroyd, Mar 17 2020

Extensions

a(1)=0 prepended and a(12)-a(18) from Andrew Howroyd, Mar 17 2020

A129418 Number of isomorphism classes of 4-regular loopless multigraphs of order n.

Original entry on oeis.org

1, 0, 1, 1, 4, 7, 24, 60, 240, 930, 4701, 26637, 178569, 1339529, 11187064, 101871881, 1002594996, 10574095327, 118850827173, 1417140114336, 17860018997346, 237160827107408, 3309078044759285, 48396906463199522, 740331404753448181, 11821525310570525197
Offset: 0

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/
Also number of carbon allotropes satisfying the octet rule, excluding stereoisomers. - Natan Arie Consigli, Jun 06 2017

Crossrefs

Column k=4 of A333330.
Cf. A129417, A129429, A129416, A129420, A129422, A129424, A129426.

Programs

Formula

Euler transform of A129417. - Andrew Howroyd, Mar 14 2020

Extensions

a(0)-a(1) by Natan Arie Consigli, Jun 06 2017
a(18)-a(25) from Andrew Howroyd, Mar 17 2020

A129420 Number of isomorphism classes of 5-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 5, 54, 1753, 189341, 46935710, 20494522535, 14041749098602, 14155266802426836, 20061744131278672638, 38587417589460488631726, 97900485588988429336271590, 320012505326477694925887757141, 1321269556386383657509085883067690, 6775074159053505093089897813890701467
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=5 of A333330.
Cf. A129419, A129431, A129416, A129418, A129422, A129424, A129426.

Formula

Euler transform of A129419. - Andrew Howroyd, Mar 17 2020

Extensions

a(8)-a(15) from Andrew Howroyd, Mar 21 2020

A129424 Number of isomorphism classes of 7-regular loopless multigraphs of order 2n.

Original entry on oeis.org

1, 8, 271, 90913, 165134083, 861889423083, 10352781874872853, 253226977418079364874, 11542716741789258789499297, 915461016322959135074642849828
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Crossrefs

Column k=7 of A333330.
Cf. A129423, A129435, A129416, A129418, A129420, A129422, A129426.

Formula

Euler transform of A129423. - Andrew Howroyd, Mar 21 2020

Extensions

a(7)-a(10) from Andrew Howroyd, Mar 21 2020

A129433 Number of isomorphism classes of 6-regular multigraphs of order n, loops allowed.

Original entry on oeis.org

1, 4, 13, 66, 384, 3369, 40365, 675368, 14843787, 412444439, 14024069358, 570883006810, 27368160642418, 1525274952640101, 97766102550401217, 7141331463919539567, 589726897233157151109, 54669618693986578729541, 5653861386413841612952683, 648651469510725736002532451
Offset: 1

Views

Author

Brendan McKay, Apr 15 2007

Keywords

Comments

Initial terms computed using software at http://users.cecs.anu.edu.au/~bdm/nauty/

Links

Crossrefs

Column k=6 of A167625.
Cf. A129432, A129422, A129427, A129429, A129431, A129435, A129437.

Formula

a(n) = N\{S_n[S_6] * S_{3n}[S_2]\}. [Equation (5.8) of Read 1959]
Euler transform of A129432. - Andrew Howroyd, Mar 15 2020

Extensions

Using the formula, new terms a(13), Oct 05 2009, and a(14)-a(16), Nov 09 2009, were computed in MAGMA by Jason Kimberley
a(17)-a(20) from Andrew Howroyd, Mar 21 2020
Showing 1-8 of 8 results.

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

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