A014396 -id:A014396 - cp's OEIS frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
;
1;
2 1;
5 2 1;
12 8 2 1;
33 22 8 2 1;
103 72 26 8 2 1;
333 229 87 26 8 2 1;
1183 782 295 92 26 8 2 1;
4442 2760 1036 315 92 26 8 2 1;
17576 10270 3735 1129 321 92 26 8 2 1;
72810 39770 13976 4117 1154 321 92 26 8 2 1;
314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..50
George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

More terms from Sean A. Irvine, Oct 02 2011

A253186 Number of connected unlabeled loopless multigraphs with 3 vertices and n edges.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330

Offset: 0

Danny Rorabaugh, Mar 23 2015

a(n) is also the number of ways to partition n into 2 or 3 parts.
a(n) is also the dimension of linear space of three-dimensional 2n-homogeneous polynomial vector fields, which have an octahedral symmetry (for a given representation), which are solenoidal, and which are vector fields on spheres. - Giedrius Alkauskas, Sep 30 2017
Apparently a(n) = A244239(n-6) for n > 4. - Georg Fischer, Oct 09 2018
a(n) is also the number of loopless connected n-regular multigraphs with 4 nodes. - Natan Arie Consigli, Aug 09 2019
a(n) is also the number of inequivalent linear [n, k=2] binary codes without 0 columns (see A034253 for more details). - Petros Hadjicostas, Oct 02 2019
Differs from A160138 only by the offset. - R. J. Mathar, May 15 2023
From Allan Bickle, Jul 13 2025: (Start)
a(n) is the number of theta graphs with n-2 vertices, or n-1 edges.  Equivalently, the number of 2-connected graphs with n-2 vertices and n-1 edges.
A theta graph has three paths with length at least 1 identified at their endpoints.  There can at most one path with length 1.
For instance the theta graphs with 6 vertices have paths with lengths (1,2,4), (1,3,3), or (2,2,2), so a(6-2) = 3. (End)

			On vertex set {a, b, c}, every connected multigraph with n = 5 edges is isomorphic to a multigraph with one of the following a(5) = 4 edge multisets: {ab, ab, ab, ab, ac}, {ab, ab, ab, ac, ac}, {ab, ab, ab, ac, bc}, and {ab, ab, ac, ac, bc}.

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Giedrius Alkauskas, Projective and polynomial superflows. I, arxiv.org/1601.06570 [math.AG], 2017; see Section 5.3.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k = 2.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,2,2}.]
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; see Eq. (23).
Gordon Royle, Small Multigraphs.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A001399, A003082, A014395, A014396, A014397, A014398, A050535, A076864.
Column k = 3 of A191646 and column k = 2 of A034253.
First differences of A034198 (excepting the first term).
Cf. A213654, A213655, A213668 (theta graphs).

[Floor(n/2) + Floor((n^2 + 6)/12): n in [0..70]]; // Vincenzo Librandi, Mar 24 2015

CoefficientList[Series[- x^2 (x^3 - x - 1) / ((1 - x) (1 - x^2) (1 - x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Mar 24 2015 *)
LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 2, 3, 4}, 61] (* Robert G. Wilson v, Oct 11 2017 *)
a[n_]:=Floor[n/2] + Floor[(n^2 + 6)/12]; Array[a, 70, 0] (* Stefano Spezia, Oct 09 2018 *)

[floor(n/2) + floor((n^2 + 6)/12) for n in range(70)]

a(n) = A004526(n) + A069905(n).
a(n) = floor(n/2) + floor((n^2 + 6)/12).
G.f.: x^2*(x^3 - x - 1)/((x - 1)^2*(x^2 - 1)*(x^2 + x + 1)).

A192517 Table read by antidiagonals: T(n,k) = number of multigraphs with n vertices and k edges, with no loops allowed (n >= 1, k >= 0).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 3, 6, 4, 1, 0, 1, 1, 3, 7, 11, 5, 1, 0, 1, 1, 3, 8, 17, 18, 7, 1, 0, 1, 1, 3, 8, 21, 35, 32, 8, 1, 0, 1, 1, 3, 8, 22, 52, 76, 48, 10, 1, 0, 1, 1, 3, 8, 23, 60, 132, 149, 75, 12, 1, 0

Offset: 1

Alberto Tacchella, Jul 03 2011

Rows converge to sequence A050535, i.e. T(n,k) = A050535(k) for n >= 2k.

			Table begins:
[1,0,0,0,0,0,0,0,0,...],
[1,1,1,1,1,1,1,1,1,...],
[1,1,2,3,4,5,7,8,10,...],
[1,1,3,6,11,18,32,48,75,...],
[1,1,3,7,17,35,76,149,291,...],
[1,1,3,8,21,52,132,313,741,...],
[1,1,3,8,22,60,173,471,1303,...],
[1,1,3,8,23,64,197,588,1806,...],
...

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 171.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (terms 1..78 from Alberto Tacchella computed using nauty 2.4, terms 79..595 from Sean A. Irvine computed using cycle index method of Harary and Palmer).
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 69.

Cf. A008406, A191646, A003082 (row 4), A014395 (row 5), A014396 (row 6).

\\ See A191646 for G function.
R(n)={Mat(vectorv(n, k, concat([1], G(k, n-1))))}
{ my(A=R(10)); for(n=1, #A, for(k=1, #A, print1(A[n,k], ", "));print) } \\ Andrew Howroyd, May 14 2018

A014395 Number of multigraphs with 5 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 7, 17, 35, 76, 149, 291, 539, 974, 1691, 2874, 4730, 7620, 11986, 18485, 27944, 41550, 60744, 87527, 124338, 174403, 241650, 331153, 448987, 602853, 801943, 1057615, 1383343, 1795578, 2313595, 2960656, 3763879, 4755505, 5972927, 7460196, 9267980

Offset: 0

N. J. A. Sloane

CRC Handbook of Combinatorial Designs, 1996, p. 650.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row 5 of A192517.

Cf. A001399, A003082, A014396, A014397, A014398.

CoefficientList[Series[PairGroupIndex[SymmetricGroup[5],s]/.Table[s[i]->1/(1-x^i),{i,1,Binomial[5,2]}],{x,0,30}],x] (* Geoffrey Critzer, Oct 14 2012 *)

concat([1], G(5, 40)) \\ See A191646 for G. - Andrew Howroyd, Mar 15 2020

G.f.: (x^21 + x^20 + 5*x^19 + 8*x^18 + 14*x^17 + 22*x^16 + 32*x^15 + 40*x^14 + 39*x^13 + 47*x^12 + 36*x^11 + 36*x^10 + 25*x^9 + 21*x^8 + 12*x^7 + 11*x^6 + 4*x^5 + 4*x^4 + x^3 + x^2 - x + 1)/((x^6 - 1)*(x^5 - 1)^2*(x^4 - 1)^2*(x^3 - 1)^2*(x - 1)^3*(x + 1)).

More terms from Vladeta Jovovic, Dec 23 1999

A003082 Number of multigraphs with 4 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 6, 11, 18, 32, 48, 75, 111, 160, 224, 313, 420, 562, 738, 956, 1221, 1550, 1936, 2405, 2958, 3609, 4368, 5260, 6279, 7462, 8814, 10356, 12104, 14093, 16320, 18834, 21645, 24783, 28272, 32158, 36442, 41187, 46410, 52151, 58443, 65345, 72864

Offset: 0

N. J. A. Sloane

Also, expansion of Molien series for representation Sym^2(R^n) of the automorphism group of the lattice D_3.

CRC Handbook of Combinatorial Designs, 1996, p. 650.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.19).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..1000
Axel Kleinschmidt and Valentin Verschinin, Tetrahedral modular graph functions, arXiv:1706.01889 [hep-th], 2017, p. 20.
P. Sarnak and A. Strömbergsson, Minima of Epstein's zeta function and heights of flat tori, Inventiones mathematicae, July 2006, Volume 165, Issue 1, pp 115-151.
Index entries for linear recurrences with constant coefficients, signature (2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1).

Cf. A001399, A014395 (5 nodes), A014396, A014397, A014398, row 4 of A192517.

Cf. A290778 (connected).

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2) )); // G. C. Greubel, Nov 04 2022

CoefficientList[Series[PairGroupIndex[SymmetricGroup[4], s] /.Table[s[i] -> 1/(1 - x^i), {i, 1, 4}], {x, 0, 40}], x] (* Geoffrey Critzer, Nov 10 2011 *)
LinearRecurrence[{2,0,0,-2,-2,3,0,3,-2,-2,0,0,2,-1},{1,1,3,6,11,18,32,48,75,111, 160,224,313,420},50] (* Harvey P. Dale, Oct 09 2016 *)

Vec((x^8-x^7+x^6+x^4+x^2-x+1)/((x-1)^6*(x+1)^2*(x^2+1)*(x^2+x+1)^2) + O(x^100)) \\ Colin Barker, Apr 02 2015

def A003082_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2) ).list()
A003082_list(50) # G. C. Greubel, Nov 04 2022

G.f.: (1-x+x^2+x^4+x^6-x^7+x^8)/((1-x)^6*(1+x)^2*(1+x^2)*(1+x+x^2)^2).
a(n) = 2*a(n-1) - 2*a(n-4) - 2*a(n-5) + 3*a(n-6) + 3*a(n-8) - 2*a(n-9) - 2*a(n-10) + 2*a(n-13) - a(n-14). - Wesley Ivan Hurt, Apr 20 2021
a(n) = (1/17280)*((3 + n)*(3175 + 2088*n + 564*n^2 + 72*n^3 + 6*n^4 + 945*(-1)^n) + 540*I^n*(1 + (-1)^n)) + (1/27)*(3*ChebyshevU(n, -1/2) + 2*ChebyshevU(n-1, -1/2) + 3*(-1)^n*(A099254(n) - A099254(n-1))). - G. C. Greubel, Nov 04 2022

Entry improved by comments from Vladeta Jovovic, Dec 23 1999

A014397 Number of loopless multigraphs with 7 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 8, 22, 60, 173, 471, 1303, 3510, 9234, 23574, 58464, 140340, 326792, 738090, 1619321, 3455129, 7180856, 14555856, 28819926, 55808840, 105834657, 196779279, 359124362, 643976482, 1135731758, 1971734302, 3372477533

Offset: 0

N. J. A. Sloane

nonn

CRC Handbook of Combinatorial Designs, 1996, p. 650.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row 7 of A192517.

Cf. A001399, A003082, A014395, A014396, A014398.

concat([1], G(7, 30)) \\ See A191646 for G. - Andrew Howroyd, Mar 15 2020

More terms and better description from Vladeta Jovovic, Dec 29 1999

A014398 Number of loopless multigraphs with 8 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 8, 23, 64, 197, 588, 1806, 5509, 16677, 49505, 143761, 406091, 1114890, 2970964, 7685972, 19311709, 47170674, 112123118, 259662333, 586583731, 1294143065, 2791716176, 5895027869, 12198014683, 24758285639, 49339306519

Offset: 0

N. J. A. Sloane

nonn

CRC Handbook of Combinatorial Designs, 1996, p. 650.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row 8 of A192517.

Cf. A001399, A003082, A014395, A014396, A014397.

concat([1], G(8, 30)) \\ See A191646 for G. - Andrew Howroyd, Mar 15 2020

More terms and better description from Vladeta Jovovic, Dec 29 1999

A050910 Number of pure 3-complexes on 8 unlabeled nodes with n multiple 3-simplexes.

Original entry on oeis.org

1, 1, 5, 21, 131, 940, 7902, 69025, 594203, 4856288, 37189863, 265916174, 1778005595, 11154474602, 65921168577, 368463685296, 1955231394323, 9884229508860, 47752849848446, 221109815065563, 983764137502726

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001401, A014396.

\\ See A304942 for Blocks
a(n)={Blocks(n,8,4)} \\ Andrew Howroyd, Feb 09 2020

cp's OEIS Frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

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A253186 Number of connected unlabeled loopless multigraphs with 3 vertices and n edges.

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A192517 Table read by antidiagonals: T(n,k) = number of multigraphs with n vertices and k edges, with no loops allowed (n >= 1, k >= 0).

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A014395 Number of multigraphs with 5 nodes and n edges.

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A003082 Number of multigraphs with 4 nodes and n edges.

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A014397 Number of loopless multigraphs with 7 nodes and n edges.

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A014398 Number of loopless multigraphs with 8 nodes and n edges.

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