cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A322151 Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 2, 5, 27, 216, 2311, 30988, 499919, 9431026, 203743252, 4960335470, 134382267082, 4009794148101, 130668970606412, 4617468180528235, 175867725701333896, 7182126650899080024, 313063334893103361130, 14507460736615554141354, 712192629608088061633746
Offset: 0

Views

Author

Gus Wiseman, Nov 28 2018

Keywords

Links

Crossrefs

Row sums of A322147. The unlabeled version is A191970.
Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A291842 (planar case), A321254, A322114, A322115.

Programs

  • Mathematica
    multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
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[multsubs[Range[n+1],2],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]
  • PARI
    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+1,2)))))} \\ Andrew Howroyd, Nov 28 2018

Extensions

Terms a(7) and beyond from Andrew Howroyd, Nov 28 2018

A339071 Triangle read by rows: T(n,k) is the number of unlabeled simple nonseparable (or 2-connected) graphs with n nodes and k edges (n >= 1, n-1 <= k <= n*(n-1)/2).

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 2, 1, 1, 0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1, 0, 1, 4, 20, 50, 82, 94, 81, 59, 38, 20, 10, 5, 2, 1, 1, 0, 1, 6, 40, 161, 429, 780, 1076, 1197, 1114, 885, 622, 386, 215, 112, 55, 24, 11, 5, 2, 1, 1, 0, 1, 7, 70, 433, 1729, 4796
Offset: 1

Views

Author

Andrew Howroyd, Nov 23 2020

Keywords

Examples

			Triangle T(n,k) begins:
======================================================
n/k | 0  1  2  3  4  5  6  7  8   9  10 11 12 13 14 15
----+-------------------------------------------------
  1 | 0;
  2 |    1;
  3 |       0, 1;
  4 |          0, 1, 1, 1;
  5 |             0, 1, 2, 3, 2,  1,  1;
  6 |                0, 1, 3, 9, 14, 12, 8, 5, 2, 1, 1;
  ...

Links

Crossrefs

Row sums are A002218.
Column sums are A010355.
Cf. A054923, A054924, A123534, A339070 (transpose), A339072.

A046742 Triangle of number of connected graphs with k >= 1 edges and n nodes (2 <= n <= k+1).

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			1;
0 1;
0 1 2;
0 0 2 3;
0 0 1 5 6;
0 0 1 5 13 11;
0 0 0 4 19 33 23;
0 0 0 2 22 67 89 47;
0 0 0 1 20 107 236 240 106;
0 0 0 1 14 132 486 797 657 235;
0 0 0 0 9 138 814 2075 2678 1806 551;
0 0 0 0 5 126 1169 4495 8548 8833 5026 1301;
0 0 0 0 2 95 1454 8404 22950 33851 28908 13999 3159;
0 0 0 0 1 64 1579 13855 53863 109844 130365 93569 39260 7741;
0 0 0 0 1 40 1515 20303 112618 313670 499888 489387 300748 110381 19320;
0 0 0 0 0 21 1290 26631 211866 803905 1694642 2179949 1799700 959374 311465 ...
... (so with 5 edges there's 1 graph with 4 nodes, 5 with 5 nodes and 1 with 6 nodes).

Links

Crossrefs

Cf. A002905 (row sums), A008406, A046751, A054923, A054924 (transpose), A001349 (column sums).

Extensions

Data corrected by Sean A. Irvine, Apr 23 2021

A046751 Triangle read by rows of number of connected graphs with n nodes and k edges (n >= 2, 1 <= k <= n(n-1)/2).

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 2, 1, 1, 0, 0, 0, 3, 5, 5, 4, 2, 1, 1, 0, 0, 0, 0, 6, 13, 19, 22, 20, 14, 9, 5, 2, 1, 1, 0, 0, 0, 0, 0, 11, 33, 67, 107, 132, 138, 126, 95, 64, 40, 21, 10, 5, 2, 1, 1, 0, 0, 0, 0, 0, 0, 23, 89, 236, 486, 814, 1169, 1454, 1579, 1515, 1290, 970, 658, 400, 220, 114
Offset: 2

Views

Author

N. J. A. Sloane

Keywords

Examples

			1;
0,1,1;
0,0,2,2,1, 1;
0,0,0,3,5, 5, 4, 2,  1,  1;
0,0,0,0,6,13,19,22, 20, 14,  9,  5, 2, 1, 1;
0,0,0,0,0,11,33,67,107,132,138,126,95,64,40,21,10,5,2,1,1;
[ the 4th row giving the numbers of connected graphs with 4 nodes and from 1 to 10 edges ].

Links

Crossrefs

See A054924, which is the main entry for this triangle.
Cf. A002905, A008406, A046742, A054923, A054924.

Extensions

More terms from Vladeta Jovovic, Apr 21 2000

A322133 Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 8, 3, 1, 0, 7, 17, 12, 3, 1, 0, 11, 46, 45, 18, 4, 1, 0, 15, 94, 141, 76, 23, 4, 1, 0, 22, 212, 432, 333, 124, 30, 5, 1, 0, 30, 416, 1231, 1254, 622, 178, 37, 5, 1, 0, 42, 848, 3346, 4601, 2914, 1058, 252, 45, 6, 1
Offset: 0

Views

Author

Gus Wiseman, Nov 27 2018

Keywords

Comments

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Triangle begins:
    1
    0    1
    0    2    1
    0    3    2    1
    0    5    8    3    1
    0    7   17   12    3    1
    0   11   46   45   18    4    1
    0   15   94  141   76   23    4    1
    0   22  212  432  333  124   30    5    1
    0   30  416 1231 1254  622  178   37    5    1
    0   42  848 3346 4601 2914 1058  252   45    6    1
Non-isomorphic representatives of the multiset partitions counted in row 4:
  {{1,1,1,1}}        {{1,1,2,2}}      {{1,2,3,3}}    {{1,2,3,4}}
  {{1},{1,1,1}}      {{1,2,2,2}}      {{1,3},{2,3}}
  {{1,1},{1,1}}      {{1},{1,2,2}}    {{3},{1,2,3}}
  {{1},{1},{1,1}}    {{1,2},{1,2}}
  {{1},{1},{1},{1}}  {{1,2},{2,2}}
                     {{2},{1,2,2}}
                     {{1},{2},{1,2}}
                     {{2},{2},{1,2}}

Links

Crossrefs

Row sums are A007718.
Cf. A000664, A007716, A054923, A191646, A191970, A275421, A286520, A317672, A319719, A321155, A321254, A322114.

Programs

  • PARI
    \\ Needs G(m,n) defined in A317533 (faster PARI).
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={[Vecrev(p) | p <- Vec(1 + InvEulerMTS(y^n*G(n,n) + sum(k=0, n-1, y^k*(1 - y)*G(k,n))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 15 2024

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1
Offset: 0

Views

Author

Gus Wiseman, Jan 19 2024

Keywords

Comments

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

Examples

			Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

Links

Crossrefs

Row lengths are A000124.
Diagonal T(n,n-1) is A000272, rooted A000169.
The case without loops is A062734.
Row sums are A062740.
Transpose is A322147.
Column sums are A322151.
Diagonal T(n,n) is A368951, connected case of A368597.
Connected case of A369199, without loops A054548.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

Programs

  • Mathematica
    csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]
  • PARI
    T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

Formula

E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

A384850 Triangle read by rows: T(n,k) is the number of unsensed simple planar maps with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 3, 0, 0, 0, 1, 7, 6, 0, 0, 0, 1, 7, 22, 12, 0, 0, 0, 0, 5, 42, 76, 27, 0, 0, 0, 0, 2, 49, 237, 271, 65, 0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175, 0, 0, 0, 0, 0, 18, 510, 3539, 6757, 3765, 490
Offset: 0

Views

Author

Andrew Howroyd, Jun 13 2025

Keywords

Comments

The planar maps considered here are connected.
The initial terms of this sequence can be computed using the tool "plantri", in particular the command "./plantri -u -v -c1 -p [n]" will compute values for a column.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 2;
  0, 0, 0, 2, 3;
  0, 0, 0, 1, 7,  6;
  0, 0, 0, 1, 7, 22,  12;
  0, 0, 0, 0, 5, 42,  76,   27;
  0, 0, 0, 0, 2, 49, 237,  271,   65;
  0, 0, 0, 0, 1, 35, 442, 1293, 1001, 175;
  ...

Links

Crossrefs

Row sums are A006395.
Column sums are A372892.
Main diagonal is A006082.
Subdiagonal is A384967.
Cf. A054923 (graphs), A277741 (not necessarily simple), A342060 (2-connected), A212438 (3-connected), A384963 (version by number of vertices then faces).

A001436 Number of connected graphs with n nodes, n+2 edges.

Original entry on oeis.org

0, 0, 0, 1, 4, 22, 107, 486, 2075, 8548, 33851, 130365, 489387, 1799700, 6499706, 23118465, 81134475, 281454170, 966388692, 3288208176, 11098235911, 37188198356, 123800999503, 409715126169, 1348690034859, 4417932007626, 14407260221164
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • 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).

Links

Crossrefs

A diagonal of A054923.

Extensions

Description corrected Aug 02 1996.
More terms from Sean A. Irvine, Jul 23 2012

A343873 Triangle read by rows: T(n,k) is the number of unlabeled connected planar graphs with n edges and k nodes (n >= 0, 1 <= k <= n + 1).

Original entry on oeis.org

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, 19, 107, 236, 240, 106, 0, 0, 0, 0, 0, 13, 130, 486, 797, 657, 235, 0, 0, 0, 0, 0, 5, 130, 804, 2075, 2678, 1806, 551
Offset: 0

Views

Author

Andrew Howroyd, May 06 2021

Keywords

Comments

First differs from A054923 in row n=9.
Terms may be computed using the tools geng and planarg in nauty.

Examples

			Triangle begins (n edges >= 0, k vertices >= 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, 19, 107, 236, 240, 106;
  0, 0, 0, 0, 0, 13, 130, 486, 797, 657, 235;
  ...

Links

Crossrefs

Row sums are A046091.
Column sums are A003094.
Main diagonal is A000055.
Subsequent diagonals are A001429, A001435, A001436 (same as for not necessarily planar graphs).
Cf. A049334 (transpose), A054923, A343870.

Programs

  • nauty
    geng -c $k $n:$n | planarg -q | countg -q # Georg Grasegger, Jul 06 2023

A094602 Total number of edges in all connected unlabeled graphs on n nodes.

Original entry on oeis.org

0, 1, 5, 25, 130, 951, 9552, 160220, 4756703, 264964172, 27746801125, 5419696866001, 1964101824992259, 1319988856541150115, 1648566523004692022634, 3838409698542815296758222, 16719797018733808721401666187, 136732968429033400292302529059213
Offset: 1

Views

Author

Vladeta Jovovic, Jun 06 2004

Keywords

Links

Crossrefs

Cf. A046751, A054923, A054924, A086314, A095351.

Programs

  • PARI
    \\ See A054923 for G, InvEulerMT.
a(n)={subst(deriv(InvEulerMT(vector(n, k, G(k,y)))[n]), y, 1)} \\ Andrew Howroyd, Feb 01 2020

Formula

a(n) = Sum_{k=1..binomial(n,2)} k*A054924(n, k). - Andrew Howroyd, Feb 01 2020

Extensions

Terms a(17) and beyond from Andrew Howroyd, Feb 01 2020
Previous Showing 11-20 of 21 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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