A059441 -id:A059441 - cp's OEIS frontend

A295193 Number of regular simple graphs on n labeled nodes.

1, 2, 2, 8, 14, 172, 932, 45936, 1084414, 155862512, 10382960972, 6939278572096, 2203360500122300, 4186526756621772344, 3747344008241368443820, 35041787059691023579970848, 156277111373303386104606663422, 4142122641757598618318165240180096

Offset: 1

Álvar Ibeas, Nov 16 2017

nonn

			From _Gus Wiseman_, Dec 19 2018: (Start)
A graph is regular if all vertices have the same degree. For example, the a(4) = 8 simple regular graphs are:
  1 2
  3 4
.
  4---1  3---1  2---1
  3---2  4---2  4---3
.
  3---4  4---3  4---2
  |   |  |   |  |   |
  1---2  1---2  1---3
.
  4---3
  | X |
  2---1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..24
E. A. Bender and E. R. Canfield, The asymptotic number of labeled graphs with given degree sequences, Journal of Combinatorial Theory, Series A, 24 (1978), 296-307.
Andrew Howroyd, PARI Program
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
B. D. McKay, Applications of a technique for labelled enumeration, Congress. Numerantium, 40 (1983), 207-221.
Wikipedia, Regular graph

Row sums of A059441.

Cf. A005176 (unlabeled equivalent), A058891, A116539, A299353, A306017, A306021, A319189, A319190, A319612, A319729.

Table[Sum[SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[n],{2}]}],Sequence@@Table[{x[i],0,k},{i,n}]],{k,0,n-1}],{n,1,9}] (* Gus Wiseman, Dec 19 2018 *)

\\ See link for program file.
for(n=1, 10, print1(A295193(n), ", ")) \\ Andrew Howroyd, Aug 28 2019

a(16)-a(18) from Andrew Howroyd, Aug 28 2019

A001205 Number of clouds with n points; number of undirected 2-regular labeled graphs; or number of n X n symmetric matrices with (0,1) entries, trace 0 and all row sums 2.

Original entry on oeis.org

1, 0, 0, 1, 3, 12, 70, 465, 3507, 30016, 286884, 3026655, 34944085, 438263364, 5933502822, 86248951243, 1339751921865, 22148051088480, 388246725873208, 7193423109763089, 140462355821628771, 2883013994348484940

Offset: 0

N. J. A. Sloane

a(n) is the number of ways of covering K_n with cycles of length >= 3. Also number of 'frames' on n lines: given n lines in general position (none parallel and no three concurrent), a frame is a subset of n of the e C(n,2) points of intersection such that no three points are on the same line. - Mitch Harris, Jul 06 2006

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 410-411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 276 and 279.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.
Ph. Flajolet, Singular combinatorics, pp. 561-571, Proc. Internat. Congr. Math., Beijing 2002, Higher Education Press, Beijing, 2002, Vol III.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.6b, 3.3.34.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.8. Also problems 5.23 and 5.15(a), case k=3.
Z. Tan and S. Gao, Enumeration of (0,1)-Symmetric Matrices, submitted [From Shanzhen Gao, Jun 05 2009] [apparently unpublished as of 2016]
H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 77, Eq. 3.9.1.
W. A. Whitworth, Choice and Chance, Bell, 1901, p. 269, ex. 160.

T. D. Noe, Table of n, a(n) for n=0..100
Editorial note, Robinson's constant, Amer. Math. Monthly, 59 (1952), 296-297.
Ph. Flajolet, Singular combinatorics, arXiv:math/0304465 [math.CO], 2003.
Ph. Flajolet and A. Odlyzko, Singularity analysis of generating functions, [Research Report] RR-0826, INRIA. 1988.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
H. Richter, Analyzing coevolutionary games with dynamic fitness landscapes, arXiv preprint arXiv:1603.06374 [q-bio.PE], 2016.
R. Robinson, A new absolute geometric constant?, Amer. Math. Monthly, 58 (1951), 462-469.
Weiping Wang, Tianming Wang, An automatic approach to the generating functions for some special sequences, Ars. Comb. 116 (2018) 263, Example 4.2
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 86, Eq. 3.9.1.

Cf. A000985, A000986, A002137. A diagonal of A059441 and A144163.

a := n -> (-1)^n*n!*add((3/4)^k*binomial(-1/2, n-k)*hypergeom([1/2,-k], [1/2-n+k], 1/3)/ k!, k=0..n): seq(simplify(a(n)), n=0..21); # Peter Luschny, Aug 26 2017

m = 21; CoefficientList[ Series[ Exp[-x/2 - x^2/4] / Sqrt[1-x], {x, 0, m}], x]*Table[n!, {n, 0, m}] (* Jean-François Alcover, Jun 21 2011, after e.g.f. *)

a(n):=sum(sum(binomial(k,i)*binomial(i-1/2,n-k)*(3^(k-i)*n!)/(4^k*k!)*(-1)^(n-i),i,0,k),k,0,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Aug 25 2017 */

a(n)=if(n<0,0,n!*polcoeff(exp(-x/2-x^2/4+x*O(x^n))/sqrt(1-x+x*O(x^n)),n))

a(n) ~ n!*exp(-3/4)/sqrt(Pi*n).
E.g.f.: exp(-x/2-x^2/4)/sqrt(1-x).
D-finite with recurrence a(n+1) = n*(a(n)+a(n-2)*(n-1)/2).
1/4^n * Sum_{b=0..floor(n/2)} Sum_{g=0..n-2*b} (-1)^(b+g) * 2^(2b+g) * n! * (2n-4b-2g)! / (b! * g! * (n-2b-g)!^2). - Shanzhen Gao, Jun 05 2009
a(n) = (-1)^n*n!*Sum_{k=0..n}(3/4)^k*binomial(-1/2, n - k)*hypergeom([1/2, -k], [1/2 - n + k], 1/3)/ k!. - Peter Luschny, Aug 26 2017

A002829 Number of trivalent (or cubic) labeled graphs with 2n nodes.

Original entry on oeis.org

1, 0, 1, 70, 19355, 11180820, 11555272575, 19506631814670, 50262958713792825, 187747837889699887800, 976273961160363172131825, 6840300875426184026353242750, 62870315446244013091262178375075, 741227949070136911068308523257857500

Offset: 0

N. J. A. Sloane

nonn

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
R. W. Robinson, Computer print-out, no date. Gives first 30 terms.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..160 (first 31 terms from R. W. Robinson)
F. Chyzak, M. Mishna and B. Salvy, Effective D-Finite Symmetric Functions, FPSAC '02 (2002) # 19.12, see Maple output prior to the References.
Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.
Oleg Evnin and Weerawit Horinouchi, A Gaussian integral that counts regular graphs, arXiv:2403.04242 [cond-mat.stat-mech], 2024. See p. 12.
I. P. Goulden and D. M. Jackson, Labelled graphs with small vertex degrees and P-recursiveness, SIAM J. Algebraic Discrete Methods 7(1986), no. 1, 60-66. MR0819706 (87k:05093)
I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence)
R. W. Robinson, Cubic labeled graphs, computer print-out, n.d.
N. C. Wormald, Enumeration of labelled graphs II: cubic graphs with a given connectivity, J. Lond Math Soc s2-20 (1979) 1-7, eq. (2.6).

A diagonal of A059441. Cf. A005814.

See A004109 for connected graphs of this type.

From R. J. Mathar, Oct 31 2010: (Start)
A002829aux := proc(i) local a,j,k ; a := 0 ; for j from 0 to i do for k from 0 to 2*(i-j) do a := a+(-1)^(j+k)/j!*doublefactorial(2*i+2*k-1)/3^k/k!/(2*i-2*j-k)! ; end do: end do: a*3^i/2^i ; end proc:
A002829 := proc(n) (2*n)!/6^n*add( A002829aux(i)/(n-i)!,i=0..n) ; end proc: seq(A002829(n),n=0..6) ; (End)
egf := hypergeom([1/6, 5/6],[],12*x/(x^2+8*x+4)^(3/2)) * exp(-ln(1/4*x^2+2*x+1)/4 - x/3 + (x^2+8*x+4)^(3/2)/(24*x) - 1/(3*x) - x^2/24 - 1):
ser := convert(series(egf,x=0,30),polynom):
seq(coeff(ser,x,i) * (2*i)!, i=0..degree(ser)); # Mark van Hoeij, Nov 07 2011

Flatten[{1,RecurrenceTable[{2 (-3+n) (-2+n) (-1+n) (-7+2 n) (-5+2 n) (-3+2 n) (-1+2 n) (-4+3 n) (-1+3 n) a[-4+n]-2 (-2+n) (-1+n) (-5+2 n) (-3+2 n) (-1+2 n) (-1+3 n) (43-42 n+9 n^2) a[-3+n]-(-1+n) (-3+2 n) (-1+2 n) (-104+501 n-441 n^2+108 n^3) a[-2+n]-9 (-1+n) (-1+2 n) (-7+3 n) (2-4 n+3 n^2) a[-1+n]+3 (-7+3 n) (-4+3 n) a[n]==0,a[1]==0,a[2]==1,a[3]==70,a[4]==19355},a,{n,1,15}]}] (* Vaclav Kotesovec, Mar 11 2014 *)
terms = 14;
egf = HypergeometricPFQ[{1/6, 5/6}, {}, 12x/(x^2 + 8x + 4)^(3/2)] Exp[-Log[ 1/4 x^2 + 2x + 1]/4 - x/3 + (x^2 + 8x + 4)^(3/2)/(24x) - 1/(3x) - x^2/24 - 1] + O[x]^terms;
CoefficientList[egf, x] (2 Range[0, terms-1])! (* Jean-François Alcover, Nov 23 2018, after Mark van Hoeij *)

a(n) = sum(i=0, 2*n, sum(k=0, min(floor((3*n-i)/3), floor((2*n-i)/2)), sum(j=0, min(floor((3*n-i-3*k)/2), floor((2*n-i-2*k)/2)), ((-1)^(i+j)*(2*n)!*(2*(3*n-i-2*j-3*k))!)/(2^(5*n-i-2*j-4*k)*3^(2*n-i-2*j-k)*(3*n-i-2*j-3*k)!*i!*j!*k!*(2*n-i-2*j-2*k)!)))); \\ Michel Marcus, Jan 18 2018

From Vladeta Jovovic, Mar 25 2001: (Start)
E.g.f. f(x) = Sum_{n >= 0} a(2 * n) * x^n/(2 * n)! satisfies differential equation 6 * x^2 * (-x^2 - 2 * x + 2) * (d^2/dx^2)f(x) - (x^5 + 6 * x^4 + 6 * x^3 - 32 * x + 8) * (d/dx)f(x) + (x/6) * (-x^2 - 2 * x + 2)^2 * f(x) = 0.
Recurrence: a(2 * n) = (2 * n)!/n! * v(n) where 48 * v(n) + (-72 * n^2 + 24 * n + 48) * v(n - 1) + (72 * n^3 - 432 * n^2 + 788 * n - 428) * v(n - 2) + (36 * n^4 - 324 * n^3 + 1052 * n^2 - 1428 * n + 664) * v(n - 3) + (36 * n^4 - 360 * n^3 + 1260 * n^2 - 1800 * n + 864) * v(n - 4) + (6 * n^5 - 94 * n^4 + 550 * n^3 - 1490 * n^2 + 1844 * n - 816) * v(n - 5) + (-n^5 + 15 * n^4 - 85 * n^3 + 225 * n^2 - 274 * n + 120) * v(n - 6) = 0. (End)
a(n) = Sum_{i=0..2*n} Sum_{k=0..min(floor((3*n-i)/3), floor((2*n-i)/2))} Sum_{j=0..min(floor((3*n-i-3*k)/2), floor((2*n-i-2*k)/2))} ((-1)^(i+j)*(2*n)!*(2*(3*n-i-2*j-3*k))!)/(2^(5*n-i-2*j-4*k)*3^(2*n-i-2*j-k)*(3*n-i-2*j-3*k)!*i!*j!*k!*(2*n-i-2*j-2*k)!). - Shanzhen Gao, Jun 05 2009
E.g.f.: hypergeom([1/6, 5/6],[],12*x/(x^2+8*x+4)^(3/2))*exp(-log(1/4*x^2+2*x+1)/4 - x/3 + (x^2+8*x+4)^(3/2)/(24*x) - 1/(3*x) - x^2/24 - 1). Multiply x^i by (2*i)! to get the generating function. - Mark van Hoeij, Nov 07 2011
From Vaclav Kotesovec, Mar 11 2014: (Start)
D-finite with recurrence: 3*(3*n-7)*(3*n-4)*a(n) = 9*(n-1)*(2*n-1)*(3*n-7)*(3*n^2 - 4*n + 2)*a(n-1) + (n-1)*(2*n-3)*(2*n-1)*(108*n^3 - 441*n^2 + 501*n - 104)*a(n-2) + 2*(n-2)*(n-1)*(2*n-5)*(2*n-3)*(2*n-1)*(3*n-1)*(9*n^2 - 42*n + 43)*a(n-3) - 2*(n-3)*(n-2)*(n-1)*(2*n-7)*(2*n-5)*(2*n-3)*(2*n-1)*(3*n-4)*(3*n-1)*a(n-4).
a(n) ~ sqrt(2) * 6^n * n^(3*n) / exp(3*n+2). (End)

More terms from Vladeta Jovovic, Mar 25 2001

A051031 Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 2, 0, 2, 0, 1, 1, 1, 3, 6, 6, 3, 1, 1, 1, 0, 4, 0, 16, 0, 4, 0, 1, 1, 1, 5, 21, 60, 60, 21, 5, 1, 1, 1, 0, 6, 0, 266, 0, 266, 0, 6, 0, 1, 1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1, 1, 0, 10, 0, 10786, 0, 367860, 0, 10786

Offset: 1

Eric W. Weisstein

A graph in which every node has r edges is called an r-regular graph. The triangle is symmetric because if an n-node graph is r-regular, than its complement is (n - 1 - r)-regular and two graphs are isomorphic if and only if their complements are isomorphic.

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 A295193. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 08 2020

			T(8,3) = 6. Edge-lists for the 6 3-regular 8-node graphs:
  Graph 1: 12, 13, 14, 23, 24, 34, 56, 57, 58, 67, 68, 78
  Graph 2: 12, 13, 14, 24, 34, 26, 37, 56, 57, 58, 68, 78
  Graph 3: 12, 13, 23, 14, 47, 25, 58, 36, 45, 67, 68, 78
  Graph 4: 12, 13, 23, 14, 25, 36, 47, 48, 57, 58, 67, 68
  Graph 5: 12, 13, 24, 34, 15, 26, 37, 48, 56, 57, 68, 78
  Graph 6: 12, 23, 34, 45, 56, 67, 78, 18, 15, 26, 37, 48.
Triangle starts
  1;
  1, 1;
  1, 0, 1;
  1, 1, 1,  1;
  1, 0, 1,  0,    1;
  1, 1, 2,  2,    1,    1;
  1, 0, 2,  0,    2,    0,    1;
  1, 1, 3,  6,    6,    3,    1,    1;
  1, 0, 4,  0,   16,    0,    4,    0,  1;
  1, 1, 5, 21,   60,   60,   21,    5,  1, 1;
  1, 0, 6,  0,  266,    0,  266,    0,  6, 0, 1;
  1, 1, 9, 94, 1547, 7849, 7849, 1547, 94, 9, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..300 (rows 1..24, first 16 rows from Jason Kimberley)
Jason Kimberley, Not necessarily connected regular graphs (with girth at least 3)
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
Markus Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146. [_Jason Kimberley_, Nov 24 2009]
Markus Meringer, Fast Generation of Regular Graphs and Construction of Cages, Univ. Bayreuth, 1999
Markus Meringer, Tables of Regular Graphs
Eric Weisstein's World of Mathematics, Regular Graph.

Row sums give A005176.
Regular graphs of degree k: A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).
Cf. A059441, A068933, A068934, A295193.

T(n,r) = A068934(n,r) + A068933(n,r).

More terms and comments from David Wasserman, Feb 22 2002
More terms from Eric W. Weisstein, Oct 19 2002
Description corrected (changed 'orders' to 'degrees') by Jason Kimberley, Sep 06 2009
Extended to the sixteenth row (in the b-file) by Jason Kimberley, Sep 24 2009

A333157 Triangle read by rows: T(n,k) is the number of n X n symmetric binary matrices with k ones in every row and column.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 10, 18, 10, 1, 1, 26, 112, 112, 26, 1, 1, 76, 820, 1760, 820, 76, 1, 1, 232, 6912, 35150, 35150, 6912, 232, 1, 1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1, 1, 2620, 708256, 24243520, 133948836, 133948836, 24243520, 708256, 2620, 1

Offset: 0

Andrew Howroyd, Mar 09 2020

T(n,k) is the number of k-regular symmetric relations on n labeled nodes.
T(n,k) is the number of k-regular graphs with half-edges on n labeled vertices.
Terms may be computed without generating all graphs by enumerating the number of graphs by degree sequence. A PARI program showing this technique is given below. Burnside's lemma as applied in A122082 and A000666 can be used to extend this method to the case of unlabeled vertices A333159 and A333161 respectively.

			Triangle begins:
  1,
  1,   1;
  1,   2,     1;
  1,   4,     4,      1;
  1,  10,    18,     10,       1;
  1,  26,   112,    112,      26,      1;
  1,  76,   820,   1760,     820,     76,     1;
  1, 232,  6912,  35150,   35150,   6912,   232,   1;
  1, 764, 66178, 848932, 1944530, 848932, 66178, 764, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..230

Columns k=0..8 are A000012, A000085, A000986, A110040, A139670, A139671, A139673, A139674, A139675.
Row sums are A322698.
Central coefficients are A333164.
Cf. A188448 (transposed as array).
Cf. A059441, A295193, A333158, A333159, A333161.

\\ See script in A295193 for comments.
GraphsByDegreeSeq(n, limit, ok)={
  local(M=Map(Mat([x^0,1])));
  my(acc(p,v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r,p,i,q,v,e) = if(e<=limit && poldegree(q)<=limit, if(i<0, if(ok(x^e+q, r), acc(x^e+q, v)), my(t=polcoeff(p,i)); for(k=0,t,self()(r,p,i-1,(t-k+x*k)*x^i+q,binomial(t,k)*v,e+k)))));
  for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i,1]); recurse(n-k, p, poldegree(p), 0, src[i,2], 0))); Mat(M);
}
Row(n)={my(M=GraphsByDegreeSeq(n, n\2, (p,r)->poldegree(p)-valuation(p,x) <= r + 1), v=vector(n+1)); for(i=1, matsize(M)[1], my(p=M[i,1], d=poldegree(p)); v[1+d]+=M[i,2]; if(pollead(p)==n, v[2+d]+=M[i,2])); for(i=1, #v\2, v[#v+1-i]=v[i]); v}
for(n=0, 8, print(Row(n))) \\ Andrew Howroyd, Mar 14 2020

T(n,k) = T(n,n-k).

A005815 Number of 4-valent labeled graphs with n nodes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 465, 19355, 1024380, 66462606, 5188453830, 480413921130, 52113376310985, 6551246596501035, 945313907253606891, 155243722248524067795, 28797220460586826422720

Offset: 0

Simon Plouffe

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 411.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..260 (first 100 terms from Vincenzo Librandi)
Élie de Panafieu, Asymptotic expansion of regular and connected regular graphs, arXiv:2408.12459 [math.CO], 2024. See p. 9.
I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
Atabey Kaygun, Enumerating Labeled Graphs that Realize a Fixed Degree Sequence, arXiv:2101.02299 [math.CO], 2021.
Vaclav Kotesovec, Recurrence (of order 10)
R. C. Read and N. C. Wormald, Number of labeled 4-regular graphs, J. Graph Theory, 4 (1980), 203-212.

Cf. A005814, A002829, A005816, A272905 (connected). A diagonal of A059441.

egf := (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16));
ser := convert(series(egf,x=0,40),polynom):
seq(coeff(ser,x,i)*i!, i=0..degree(ser)); # Mark van Hoeij, Nov 07 2011

max = 17; f[x_] := HypergeometricPFQ[{1/4, 3/4}, {}, -12*x*(x + 2)*(x - 1)/(x^3 + 2*x^2 - 6*x - 6)^2]*Exp[-x*(x^2 - 6)/(8*x + 16)]/(1 + x - x^2/3 - x^3/6)^ (1/2); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Jun 19 2012, from e.g.f. *)

From Vladeta Jovovic, Mar 26 2001: (Start)
E.g.f. f(x) = Sum_{n >= 0} a(n)*x^n/(n)! satisfies the differential equation 16*x^2*(x - 1)^2*(x + 2)^2*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)*(d^2/dx^2)y(x) - 4*(x^13 + 4*x^12 - 16*x^10 - 10*x^9 - 36*x^8 - 220*x^7 - 348*x^6 - 48*x^5 + 200*x^4 - 336*x^3 - 240*x^2 + 416*x - 96)*(d/dx)y(x) - x^4*(x^5 + 2*x^4 + 2*x^2 + 8*x - 4)^2*y(x) = 0.
Recurrence: a(n) = - 1/384*(( - 256*n^2 - 896*n + 1152)*a(n - 1) + (768*n^3 - 3648*n^2 + 5568*n - 2688)*a(n - 2) + ( - 192*n^4 + 3264*n^3 - 14784*n^2 + 24384*n - 12672)*a(n - 3) + (224*n^6 - 4512*n^5 + 36304*n^4 - 148160*n^3 + 320016*n^2 - 341728*n + 137856)*a(n - 5) + ( - 640*n^5 + 8800*n^4 - 46400*n^3 + 116000*n^2 - 135360*n + 57600)*a(n - 4) + ( - 24*n^10 + 1320*n^9 - 31680*n^8 + 435600*n^7 - 3786552*n^6 + 21649320*n^5 - 82006320*n^4 + 201828000*n^3 - 306085824*n^2 + 255087360*n - 87091200)*a(n - 11) + (64*n^10 - 3480*n^9 + 82692*n^8 - 1127232*n^7 + 9726024*n^6 - 55255032*n^5 + 208179908*n^4 - 510068208*n^3 + 770738352*n^2 - 640484928*n + 218211840)*a(n - 9) + (16*n^11 - 992*n^10 + 27256*n^9 - 437160*n^8 + 4536288*n^7 - 31876656*n^6 + 154182488*n^5 - 510784360*n^4 + 1128552896*n^3 - 1570313952*n^2 + 1223830656*n - 397716480)*a(n - 10) + ( - 128*n^8 + 5488*n^7 - 94576*n^6 + 864976*n^5 - 4606672*n^4 + 14604352*n^3 - 26753984*n^2 + 25611264*n - 9630720)*a(n - 7) + (16*n^9 - 576*n^8 + 8704*n^7 - 71680*n^6 + 348880*n^5 - 1013824*n^4 + 1673376*n^3 - 1333120*n^2 + 226944*n + 161280)*a(n - 8) + (128*n^7 - 2192*n^6 + 12048*n^5 - 8240*n^4 - 151248*n^3 + 565312*n^2 - 765248*n + 349440)*a(n - 6) + ( - 4*n^13 + 364*n^12 - 14924*n^11 + 364364*n^10 - 5897892*n^9 + 66678612*n^8 - 540145892*n^7 + 3163772612*n^6 - 13344475144*n^5 + 39830815024*n^4 - 81255012384*n^3 + 106386868224*n^2 - 79211036160*n + 24908083200)*a(n - 14) + ( - 4*n^13 + 360*n^12 - 14612*n^11 + 353496*n^10 - 5674812*n^9 + 63680760*n^8 - 512439356*n^7 + 2983811688*n^6 - 12520194544*n^5 + 37201987680*n^4 - 75598952832*n^3 + 98660630016*n^2 - 73265264640*n + 22992076800)*a(n - 13) + ( - 16*n^12 + 1244*n^11 - 43208*n^10 + 884620*n^9 - 11860728*n^8 + 109396452*n^7 - 709293464*n^6 + 3243764260*n^5 - 10331326456*n^4 + 22203205904*n^3 - 30301280928*n^2 + 23300910720*n - 7504358400)*a(n - 12) + ( - n^14 + 105*n^13 - 5005*n^12 + 143325*n^11 - 2749747*n^10 + 37312275*n^9 - 368411615*n^8 + 2681453775*n^7 - 14409322928*n^6 + 56663366760*n^5 - 159721605680*n^4 + 310989260400*n^3 - 392156797824*n^2 + 283465647360*n - 87178291200)*a(n - 15)). (End)
a(n) = Sum_{d=0..floor(n/2), c=0..floor(n/2-d), b=0..(n-2c-2d), f=0..(n-2c-2d-b), k=0..min(n-b-2c-2d-f, 2n-2f-2b-3c-4d), j=0..floor(k/2+f)} ((-1)^(k+2f-j+d)*n!*(k+2f)!(2(2n-k-2f-2b-3c-4d))!) / (2^(5n-2k-2f-3b-8c-7d) * 3^(n-b-c-2d-k-f)*(2n-k-2f-2b-3c-4d)!*(k+2f-2j)!*j!*b!*c!*d!*k!*f!*(n-b-2c-2d-k-f)!). - Shanzhen Gao, Jun 05 2009
E.g.f.: (1+x-(1/3)*x^2-(1/6)*x^3)^(-1/2)*hypergeom([1/4, 3/4],[],-12*x*(x+2)*(x-1)/(x^3+2*x^2-6*x-6)^2)*exp(-x*(x^2-6)/(8*x+16)). - Mark van Hoeij, Nov 07 2011
a(n) ~ n^(2*n) * 2^(n+1/2) / (3^n * exp(2*n+15/4)). - Vaclav Kotesovec, Mar 11 2014

More terms from Vladeta Jovovic, Mar 26 2001

A333351 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n labeled 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, 3, 1, 1, 0, 1, 0, 6, 0, 1, 1, 0, 1, 1, 10, 22, 15, 1, 1, 0, 1, 0, 15, 0, 130, 0, 1, 1, 0, 1, 1, 21, 158, 760, 822, 105, 1, 1, 0, 1, 0, 28, 0, 3355, 0, 6202, 0, 1, 1, 0, 1, 1, 36, 654, 12043, 93708, 190050, 52552, 945, 1, 1, 0, 1, 0, 45, 0, 36935, 0, 3535448, 0, 499194, 0, 1

Offset: 0

Andrew Howroyd, Mar 15 2020

			Array begins:
=================================================================
n\k | 0   1    2      3       4        5         6          7
----+------------------------------------------------------------
  0 | 1   1    1      1       1        1         1          1 ...
  1 | 1   0    0      0       0        0         0          0 ...
  2 | 1   1    1      1       1        1         1          1 ...
  3 | 1   0    1      0       1        0         1          0 ...
  4 | 1   3    6     10      15       21        28         36 ...
  5 | 1   0   22      0     158        0       654          0 ...
  6 | 1  15  130    760    3355    12043     36935     100135 ...
  7 | 1   0  822      0   93708        0   3226107          0 ...
  8 | 1 105 6202 190050 3535448 45163496 431400774 3270643750 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..405

Rows n=4..6 are A000217(n+1), A244868 (with interspersed zeros), A244878.
Columns k=0..4 are A000012, A123023, A002137, A108243 (with interspersed zeros), A367497.
Cf. A059441 (graphs), A333157, A333330 (unlabeled nodes), A333467 (with loops).

MultigraphsByDegreeSeq(n, limit, ok)={
  local(M=Map(Mat([0, 1])));
  my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
  my(recurse(r, h, p, q, v, e) = if(!p, if(ok(x^e+q, r), acc(x^e+q, v)), my(i=poldegree(p), t=pollead(p)); self()(r, limit, p-t*x^i, q+t*x^i, v, e); for(m=1, h-i, for(k=1, min(t, (limit-e)\m), self()(r, if(k==t, limit, i+m-1), p-k*x^i, q+k*x^(i+m), binomial(t, k)*v, e+k*m)))));
  for(r=1, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], recurse(n-r, limit, src[i, 1], 0, src[i, 2], 0))); Mat(M);
}
T(n,k)={if((n%2&&k%2)||(n==1&&k>0), 0, vecsum(MultigraphsByDegreeSeq(n, k, (p,r)->subst(deriv(p), x, 1)>=(n-2*r)*k)[,2]))}
{ for(n=0, 8, for(k=0, 7, print1(T(n,k), ", ")); print) }

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

A319729 Regular triangle read by rows where T(n,k) is the number of labeled simple graphs on n vertices where all non-isolated vertices have degree k.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 7, 1, 1, 25, 37, 5, 1, 1, 75, 207, 85, 21, 1, 1, 231, 1347, 525, 591, 7, 1, 1, 763, 10125, 21385, 23551, 3535, 113, 1, 1, 2619, 86173, 180201, 1216701, 31647, 30997, 9, 1, 1, 9495, 819133, 12066705, 77636583, 66620631, 11485825, 286929, 955, 1

Offset: 1

Gus Wiseman, Dec 17 2018

			Triangle begins:
  1
  1       1
  1       3       1
  1       9       7       1
  1      25      37       5       1
  1      75     207      85      21       1
  1     231    1347     525     591       7       1
  1     763   10125   21385   23551    3535     113       1
  1    2619   86173  180201 1216701   31647   30997       9       1

Andrew Howroyd, Table of n, a(n) for n = 1..300

Row sums are A322555.

Cf. A000569, A005176, A058891, A059441, A295193, A301481, A306017, A306019, A306021, A319169, A319190, A319612.

Table[If[k==0,1,Sum[Binomial[n,sup]*SeriesCoefficient[Product[1+Times@@x/@s,{s,Subsets[Range[sup],{2}]}],Sequence@@Table[{x[i],0,k},{i,sup}]],{sup,n}]],{n,8},{k,0,n-1}]

T(n,k) = Sum_{i=1..n} binomial(n,i)*A059441(i,k) for k > 0. - Andrew Howroyd, Dec 26 2020

A322698 Number of regular graphs with half-edges on n labeled vertices.

Original entry on oeis.org

1, 2, 4, 10, 40, 278, 3554, 84590, 3776280, 317806466, 50710452574, 15414839551538, 8964708979273634, 10008446308186072290, 21518891146915893435358, 89320970210116481106835986, 717558285660687970023516336792, 11176382741327158622885664697124082, 338202509574712032788035618665293979610

Offset: 0

Gus Wiseman, Dec 23 2018

nonn

A graph is regular if all vertices have the same degree. A half-edge is like a loop except it only adds 1 to the degree of its vertex.

			The a(3) = 10 edge sets:
  {}
  {{1},{2,3}}
  {{3},{1,2}}
  {{2},{1,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}
  {{1},{3},{1,2},{2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}

Wikipedia, Regular graph

Row sums of A333157.

Cf. A058891, A059441, A116539, A283877, A295193, A319189, A319190, A319612, A319729.

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

for(n=1, 10, print1(A322698(n), ", ")) \\ See A295193 for script, Andrew Howroyd, Aug 28 2019

a(10)-a(18) from Andrew Howroyd, Aug 28 2019

cp's OEIS Frontend

A295193 Number of regular simple graphs on n labeled nodes.

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A001205 Number of clouds with n points; number of undirected 2-regular labeled graphs; or number of n X n symmetric matrices with (0,1) entries, trace 0 and all row sums 2.

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A002829 Number of trivalent (or cubic) labeled graphs with 2n nodes.

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A051031 Triangle read by rows: T(n,r) is the number of not necessarily connected r-regular graphs with n nodes, 0 <= r < n.

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A333157 Triangle read by rows: T(n,k) is the number of n X n symmetric binary matrices with k ones in every row and column.

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A005815 Number of 4-valent labeled graphs with n nodes.

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A333351 Array read by antidiagonals: T(n,k) is the number of k-regular loopless multigraphs on n labeled nodes, n >= 0, k >= 0.

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