A191646 -id:A191646 - cp's OEIS frontend

A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 5, 6, 0, 0, 0, 1, 5, 13, 11, 0, 0, 0, 0, 4, 19, 33, 23, 0, 0, 0, 0, 2, 22, 67, 89, 47, 0, 0, 0, 0, 1, 20, 107, 236, 240, 106, 0, 0, 0, 0, 1, 14, 132, 486, 797, 657, 235, 0, 0, 0, 0, 0, 9, 138, 814, 2075, 2678, 1806, 551, 0, 0, 0, 0, 0, 5, 126, 1169, 4495, 8548, 8833, 5026, 1301

Offset: 0

N. J. A. Sloane

The diagonal n = k+1 is A000055(n). - Jonathan Vos Post, Aug 10 2008

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 5   6;
  0, 0, 0, 1, 5, 13,  11;
  0, 0, 0, 0, 4, 19,  33,  23;
  0, 0, 0, 0, 2, 22,  67,  89,  47;
  0, 0, 0, 0, 1, 20, 107, 236, 240, 106;
  ... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 6 with 6 nodes). [Typo corrected by Anders Haglund, Jul 08 2008]

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 93, Table 4.2.2; p. 241, Table A2.

Andrew Howroyd, Table of n, a(n) for n = 0..1325
G. A. Baker et al., High-temperature expansions for the spin-1/2 Heisenberg model, Phys. Rev., 164 (1967), 800-817.
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017), Table 57.
Gordon Royle, Small graphs
Gus Wiseman, Non-isomorphic representatives of the 12 connected graphs counted in row 5.

Main diagonal is A000055.
Subsequent diagonals give the number of connected unlabeled graphs with n nodes and n+k edges for k=0..2: A001429, A001435, A001436.
Cf. A002905 (row sums), A001349 (column sums), A008406, A046751 (transpose), A054924 (transpose), A046742 (w/o left column), A343088 (labeled).
Cf. A000664, A007718, A050535, A054923, A191646, A191970, A317672, A322114, A322151.

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 23 2019

a(83)-a(89) corrected by Andrew Howroyd, Oct 24 2019

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)).

A322114 Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 1, 3, 2, 0, 0, 3, 6, 3, 0, 0, 2, 11, 14, 6, 0, 0, 1, 13, 35, 33, 11, 0, 0, 0, 10, 61, 112, 81, 23, 0, 0, 0, 5, 75, 262, 347, 204, 47, 0, 0, 0, 2, 68, 463, 1059, 1085, 526, 106, 0, 0, 0, 1, 49, 625, 2458, 4091, 3348, 1376, 235

Offset: 0

Gus Wiseman, Nov 26 2018

			Triangle begins:
   1
   1   1
   0   1   1
   0   1   3   2
   0   0   3   6   3
   0   0   2  11  14   6
   0   0   1  13  35  33  11
Non-isomorphic representatives of the graphs counted in row 4:
  {{2}{3}{12}{13}}   {{4}{12}{23}{34}}   {{13}{24}{35}{45}}
  {{2}{3}{13}{23}}   {{4}{13}{23}{34}}   {{14}{25}{35}{45}}
  {{3}{12}{13}{23}}  {{4}{13}{24}{34}}   {{15}{25}{35}{45}}
                     {{4}{14}{24}{34}}
                     {{12}{13}{24}{34}}
                     {{14}{23}{24}{34}}

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Row sums are A191970. Last column is A000055.

Cf. A000664, A007716, A007718, A007719, A054923, A191646, A275421, A317533, A321254.

InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)*edges(p,i->1+x^i)); s/n!}
T(n)={Mat([Col(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])}
{my(A=T(10)); for(n=1, #A, print(A[n,1..n]))} \\ Andrew Howroyd, Oct 22 2019

Terms a(28) and beyond from Andrew Howroyd, Oct 22 2019

A076864 Number of connected loopless multigraphs with n edges.

Original entry on oeis.org

1, 1, 2, 5, 12, 33, 103, 333, 1183, 4442, 17576, 72810, 314595, 1410139, 6541959, 31322474, 154468852, 783240943, 4077445511, 21765312779, 118999764062, 665739100725, 3807640240209, 22246105114743, 132672322938379, 807126762251748

Offset: 0

N. J. A. Sloane, Nov 23 2002

Inverse Euler transform of A050535.

Andrew Howroyd, Table of n, a(n) for n = 0..50
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.
N. J. A. Sloane, Transforms
Gus Wiseman, Non-isomorphic representatives of the unlabeled connected multigraphs counted by the first 5 terms

Row sums of A191646.
Cf. A050535 (multisets), A076865, A076866, A076867.
Cf. A000664, A007718, A054923, A191970, A275421, A317672, A322114, A322137, A322148.

A050535 = Cases[Import["https://oeis.org/A050535/b050535.txt", "Table"], {, }][[All, 2]];
(* EulerInvTransform is defined in A022562 *)
Join[{1}, EulerInvTransform[A050535 // Rest]] (* Jean-François Alcover, Feb 11 2020, updated Mar 17 2020 *)

More terms from Sean A. Irvine, Oct 02 2011
Name and comment swapped by Gus Wiseman, Nov 28 2018
a(0)=1 prepended by Andrew Howroyd, Oct 23 2019

A191970 Number of connected graphs with n edges with loops allowed.

Original entry on oeis.org

1, 2, 2, 6, 12, 33, 93, 287, 940, 3309, 12183, 47133, 190061, 796405, 3456405, 15501183, 71681170, 341209173, 1669411182, 8384579797, 43180474608, 227797465130, 1229915324579, 6790642656907, 38311482445514, 220712337683628, 1297542216770482, 7779452884747298

Offset: 0

Alberto Tacchella, Jun 20 2011

nonn

Inverse Euler transform of A053419.
From R. J. Mathar, Jul 25 2017: (Start)
The Multiset Transform gives the number of graphs with n edges (loops allowed) and k components (0<=k<=n):
 1
2
2 3
6 4 4
12 15 6 5
33 36 24 8 6
93 111 64 33 10 7
287 324 207 92 42 12 8
940 1036 633 308 120 51 14 9
3309 3408 2084 966 409 148 60 16 10
12183 11897 6959 3243 1305 510 176 69 18 11
47133 43137 24415 10970 4432 1644 611 204 78 20 12
190061 163608 88402 38763 15125 5628 1983 712 232 87 22 13
796405 644905 332979 140671 53732 19316 6824 2322 813 260 96 24 14
3456405 2639871 1299054 529179 195517 68878 23515 8020 2661 914 288 105 26 15 (End)

			a(1)=2: Either one node with the edge equal to a loop, or two nodes connected by the edge. a(2)=2: Either three nodes on a chain connected by the two edges, or two nodes connected by an edge, one node with a loop. Apparently multi-loops are not allowed (?). - _R. J. Mathar_, Jul 25 2017

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 connected graphs with loops.

Row sums of A322114.

Cf. A000664, A002905, A007718, A050535, A053419, A054923, A191646, A191970, A275421, A322133, A322151, A322152.

\\ See A322114 for InvEulerMT, G.
seq(n)={vecsum([Vec(p+O(y^n), -n) | p<-InvEulerMT(vector(n, k, G(k, y + O(y^n))))])} \\ Andrew Howroyd, Oct 22 2019

Terms a(25) and beyond from Andrew Howroyd, Oct 22 2019

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

A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 6, 11, 9, 3, 1, 9, 25, 34, 20, 6, 1, 12, 52, 104, 99, 49, 11, 1, 16, 94, 274, 387, 298, 118, 23, 1, 20, 162, 645, 1295, 1428, 881, 300, 47, 1, 25, 263, 1399, 3809, 5803, 5088, 2643, 765, 106, 1, 30, 407, 2823, 10187, 20645, 24606, 17872, 7878, 1998, 235

Offset: 0

Gus Wiseman, Nov 26 2018

			Triangle begins:
  1
  1   1
  1   2   1
  1   4   4   2
  1   6  11   9   3
  1   9  25  34  20   6
  1  12  52 104  99  49  11

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A007719. Diagonal k = n-1 is A000055.

Cf. A000664, A007716, A007718, A191646, A191970, A275421, A317533, A322114.

EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)}
InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v,x)={sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(v[i]*v[j]/g))) + sum(i=1, #v, my(t=v[i]); ((t+1)\2)*x^t + if(t%2, 0, x^(t/2)))}
G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!}
R(n)={Mat(apply(p->Col(p+O(y^n), -n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))}
{ my(T=R(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Nov 30 2018

Terms a(28) and beyond from Andrew Howroyd, Nov 30 2018

A014396 Number of loopless multigraphs with 6 nodes and n edges.

Original entry on oeis.org

1, 1, 3, 8, 21, 52, 132, 313, 741, 1684, 3711, 7895, 16310, 32604, 63363, 119745, 220546, 396428, 696750, 1198812, 2022503, 3349574, 5452496, 8732932, 13776366, 21423968, 32872642, 49804323, 74560913, 110369469, 161639227

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).
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 6 of A192517.

Cf. A001399, A003082, A014395, A014397, A014398.

CoefficientList[Series[PairGroupIndex[SymmetricGroup[6],s]/.Table[s[i]->1/(1-x^i),{i,1,Binomial[6,2]}],{x,0,30}],x] (* Geoffrey Critzer, Oct 14 2012 *)

concat([1], G(6, 40)) \\ See A191646 for G. - Andrew Howroyd, Mar 15 2020

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

A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525780, 3384988809, 91976158434, 2751122721402, 89833276321440, 3179852538140115, 121287919647418118, 4959343701136929850, 216406753768138678671, 10037782414506891597734, 493175891246093032826160

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.
Gus Wiseman, The a(4) = 140 labeled connected graphs with 4 edges.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n+1],{2}],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,6}]

Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(8) and beyond from Andrew Howroyd, Nov 28 2018

cp's OEIS Frontend

A054923 Triangle read by rows: number of connected graphs with k >= 0 edges and n nodes (1<=n<=k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A322114 Regular triangle read by rows where T(n,k) is the number of unlabeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A076864 Number of connected loopless multigraphs with n edges.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A191970 Number of connected graphs with n edges with loops allowed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

A014395 Number of multigraphs with 5 nodes and n edges.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions