A005176 -id:A005176 - cp's OEIS frontend

A322706 Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 12, 12, 1, 0, 1, 70, 330, 70, 1, 0, 1, 465, 11205, 11205, 465, 1, 0, 1, 3507, 505505, 2531200, 505505, 3507, 1, 0

Offset: 1

Gus Wiseman, Dec 23 2018

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

			Triangle begins:
  1
  1       0
  1       1       0
  1       3       1       0
  1      12      12       1       0
  1      70     330      70       1       0
  1     465   11205   11205     465       1       0
  1    3507  505505 2531200  505505    3507       1       0
Row 4 counts the following hypergraphs:
  {{1}{2}{3}{4}}  {{12}{13}{24}{34}}  {{123}{124}{134}{234}}
                  {{12}{14}{23}{34}}
                  {{13}{14}{23}{24}}

Row sums are A322705. Second column is A001205. Third column is A110101.

Cf. A005176, A058891, A059441, A295193, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Table[Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{k}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,1,n}],{n,1,6}]

A382021 Number of distinct degree sequences among all simple graphs with n vertices whose degrees are consecutive integers.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 50, 118, 272, 614, 1368, 3014

Offset: 0

John P. McSorley, Mar 12 2025

A sequence of integers is consecutive if its distinct entries are consecutive integers, and a graphic sequence is a sequence of integers that can be the degree sequence of some graph. Thus a(n) is the number of consecutive graphic sequences of length n.

			For n = 5 there are 34 non-isomorphic graphs G on 5 vertices, and 24 of these have a consecutive degree sequence. However consecutive degree sequences 11222, 12223, and 22233 each correspond to 2 non-isomorphic graphs. Thus there are 21 distinct consecutive graphic sequences of length 5, and so a(5)=21.

R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford University Press (1999).

Cf. A381586, A000088, A004251, A005176.

a(11) from Sean A. Irvine, Mar 18 2025

A054916 Number of connected unlabeled regular graphs with n nodes such that complement is also connected.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 2, 12, 18, 158, 532, 18956, 389418, 50314722, 2942198334, 1698517035792, 442786966113484, 649978211591577760, 429712868499646362046, 2886054228478618206288948, 8835589045148342277740379344, 152929279364927228928017067050204, 1207932509391069805495173186013097090, 99162609848561525198669168626676490270856

Offset: 1

N. J. A. Sloane, May 23 2000

V. A. Liskovets, Some easily derivable sequences, J. Integer Sequences, 3 (2000), #00.2.2.

Cf. A005176, A005177, A068932.

a(n) = 2*A005177(n)-A005176(n) = A005177(n)-A068932(n).

Terms a(11)-a(16) appended, journal link changed to article link, and second formula included, by Jason Kimberley, Oct 24 2009

a(17)-a(24) from Andrew Howroyd, May 19 2020

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

N. J. A. Sloane, Mar 19 2017

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

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties.
Peter Steinbach, Field Guide to Simple Graphs, Volume 1, Part 17 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Hamiltonian Graph
Eric Weisstein's World of Mathematics, LCF Notation
Eric Weisstein's World of Mathematics, Regular Graph

Cf. A005176, A005177.

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

cp's OEIS Frontend

A322706 Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A382021 Number of distinct degree sequences among all simple graphs with n vertices whose degrees are consecutive integers.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Extensions

A054916 Number of connected unlabeled regular graphs with n nodes such that complement is also connected.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A283825 Number of Hamiltonian regular graphs on n nodes.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions