A322659 -id:A322659 - cp's OEIS frontend

A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

1, 1, 1, 1, 0, 1, 1, 3, 3, 1, 1, 0, 12, 0, 1, 1, 15, 70, 70, 15, 1, 1, 0, 465, 0, 465, 0, 1, 1, 105, 3507, 19355, 19355, 3507, 105, 1, 1, 0, 30016, 0, 1024380, 0, 30016, 0, 1, 1, 945, 286884, 11180820, 66462606, 66462606, 11180820, 286884, 945, 1

Offset: 1

N. J. A. Sloane, Feb 01 2001

			1;
1,   1;
1,   0,       1;
1,   3,       3,        1;
1,   0,      12,        0,          1;
1,  15,      70,       70,         15,    1;
1,   0,     465,        0,        465,    0,   1;
1, 105,    3507,    19355,      19355, 3507, 105, 1;
1,   0,   30016,        0,    1024380, ...;
1, 945,  286884, 11180820,   66462606, ...;
1,   0, 3026655,        0, 5188453830, ...;

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24)
Denis S. Krotov, [[2,10],[6,6]]-equitable partitions of the 12-cube, arXiv:2012.00038 [math.CO], 2020.
Brendan D. McKay, Applications of a technique for labeled enumeration, Congress. Numerantium, 40 (1983), 207-221. See page 216.
Wikipedia, Regular graph

Row sums are A295193.
Columns: A123023 (k=1), A001205 (k=2), A002829 (k=3, with alternating zeros), A005815 (k=4), A338978 (k=5, with alternating zeros), A339847 (k=6).
Cf. A051031 (unlabeled case), A324163 (connected case), A333351 (multigraphs).
Cf. A001147, A058891, A319189, A319190, A319612, A319729, A322635, A322659, A322698, A322704.

Table[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{n,9},{k,0,n-1}] (* Gus Wiseman, Dec 24 2018 *)

for(n=1, 10, print(A059441(n))) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(37)-a(55) from Andrew Howroyd, Aug 25 2017

A322635 Number of regular graphs with loops on n labeled vertices.

Original entry on oeis.org

2, 4, 4, 24, 78, 1908, 23368, 1961200, 75942758, 25703384940, 4184912454930, 4462909435830552, 2245354417775573206, 10567193418810168583576, 24001585002447984453495392, 348615956932626441906675011568, 2412972383955442904868321667433106, 162906453913051798826796439651249753404

Offset: 1

Gus Wiseman, Dec 21 2018

nonn

A graph is regular if all vertices have the same degree. A loop adds 2 to the degree of its vertex.

Wikipedia, Regular graph
Gus Wiseman, The a(4) = 24 regular graphs with loops.
Gus Wiseman, The a(5) = 78 regular graphs with loops.

Row sums of A333158.

Cf. A000666, A054921, A059441, A295193, A299353, A306017, A306021, A319189, A319190, A319612, A322659, A322661.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Select[Tuples[Range[n],2],OrderedQ]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,2n}],{n,6}]

for(n=1, 10, print1(A322635(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(11)-a(18) from Andrew Howroyd, Aug 28 2019

A324163 Triangle read by rows: T(n,k) is the number of connected k-regular simple graphs on n labeled vertices, (0 <= k < n).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 3, 1, 0, 0, 12, 0, 1, 0, 0, 60, 70, 15, 1, 0, 0, 360, 0, 465, 0, 1, 0, 0, 2520, 19320, 19355, 3507, 105, 1, 0, 0, 20160, 0, 1024380, 0, 30016, 0, 1, 0, 0, 181440, 11166120, 66462480, 66462606, 11180820, 286884, 945, 1

Offset: 1

Andrew Howroyd, Sep 02 2019

			Triangle begins:
  1;
  0, 1;
  0, 0,     1;
  0, 0,     3,     1;
  0, 0,    12,     0,       1;
  0, 0,    60,    70,      15,    1;
  0, 0,   360,     0,     465,    0,     1;
  0, 0,  2520, 19320,   19355, 3507,   105, 1;
  0, 0, 20160,     0, 1024380,    0, 30016, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210

Column k=2 is A001710(n-1) for n >= 3.
Column k=3 is aerated A004109.
Column k=4 is A272905.
Row sums are A322659.
Cf. A059441 (not necessarily connected), A068934 (unlabeled).

Column k is the logarithmic transform of column k of A059441.

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A059441 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) giving number of regular labeled graphs with n nodes and degree k, read by rows.

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A322635 Number of regular graphs with loops on n labeled vertices.

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Mathematica

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A324163 Triangle read by rows: T(n,k) is the number of connected k-regular simple graphs on n labeled vertices, (0 <= k < n).

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