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.

Showing 1-7 of 7 results.

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

Views

Author

N. J. A. Sloane, Nov 23 2002

Keywords

Comments

Inverse Euler transform of A050535.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

A343088 Triangle read by rows: T(n,k) is the number of connected labeled graphs with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 0, 0, 1, 16, 0, 0, 0, 15, 125, 0, 0, 0, 6, 222, 1296, 0, 0, 0, 1, 205, 3660, 16807, 0, 0, 0, 0, 120, 5700, 68295, 262144, 0, 0, 0, 0, 45, 6165, 156555, 1436568, 4782969, 0, 0, 0, 0, 10, 4945, 258125, 4483360, 33779340, 100000000
Offset: 0

Views

Author

Andrew Howroyd, Apr 14 2021

Keywords

Examples

			Triangle begins:
  1;
  0, 1;
  0, 0, 3;
  0, 0, 1, 16;
  0, 0, 0, 15, 125;
  0, 0, 0,  6, 222, 1296;
  0, 0, 0,  1, 205, 3660,  16807;
  0, 0, 0,  0, 120, 5700,  68295,  262144;
  0, 0, 0,  0,  45, 6165, 156555, 1436568, 4782969;
  ...

Links

Crossrefs

Main diagonal is A000272.
Subsequent diagonals give the number of connected labeled graphs with n nodes and n+k edges for k=0..11: A057500, A061540, A061541, A061542, A061543, A096117, A061544 A096150, A096224, A182294, A182295, A182371.
Row sums are A322137.
Column sums are A001187.
Cf. A054923 (unlabeled), A062734 (transpose), A290776 (multigraphs), A322147 (loops allowed), A331437 (series-reduced).

Programs

  • Mathematica
    row[n_] := (SeriesCoefficient[#, {y, 0, n}]& /@ CoefficientList[ Log[Sum[x^k*(1+y)^Binomial[k, 2]/k!, {k, 0, n+1}]] + O[x]^(n+2), x]* Range[0, n+1]!) // Rest;
Table[row[n], {n, 0, 9}] // Flatten (* Jean-François Alcover, Aug 03 2022, after Andrew Howroyd *)
  • PARI
    Row(n)={Vec(serlaplace(polcoef(log(O(x^2*x^n)+sum(k=0, n+1, x^k*(1 + y + O(y*y^n))^binomial(k, 2)/k!)), n, y)), -(n+1))}
{ for(n=0, 8, print(Row(n))) }

A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 0, 1, 10, 16, 0, 0, 12, 79, 125, 0, 0, 6, 162, 847, 1296, 0, 0, 1, 179, 2565, 11436, 16807, 0, 0, 0, 116, 4615, 47100, 185944, 262144, 0, 0, 0, 45, 5540, 121185, 987567, 3533720, 4782969, 0, 0, 0, 10, 4720, 220075, 3376450, 23315936, 76826061, 100000000
Offset: 0

Views

Author

Gus Wiseman, Nov 28 2018

Keywords

Examples

			Triangle begins:
  1
  1     1
  0     2     3
  0     1    10    16
  0     0    12    79   125
  0     0     6   162   847  1296
  0     0     1   179  2565 11436 16807

Links

Crossrefs

Row sums are A322151. Last column is A000272.
Column sums are A062740.
Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.

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[If[n==0,1,Length[Select[Subsets[multsubs[Range[k],2],{n}],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,6},{k,1,n+1}]
  • 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}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, (1 + x + O(x*x^n) )^binomial(j+1,2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Extensions

Terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 0, 1, 3, 18, 131, 1180, 12570, 154535, 2151439, 33431046, 573197723, 10743619285, 218447494812, 4787255999220, 112454930390211, 2818138438707516, 75031660452368001, 2114705500316025737, 62890323682634277951, 1967901134191778583146, 64623905086814216468839
Offset: 0

Views

Author

Gus Wiseman, Nov 27 2018

Keywords

Links

Crossrefs

Cf. A002218, A013922, A123534, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

Programs

  • PARI
    seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

Formula

a(n) = Sum_{i=3..n} A123534(i, n). - Andrew Howroyd, Nov 30 2018

Extensions

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

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

A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 51, 127, 125, 1, 15, 126, 574, 1347, 1296, 1, 21, 266, 1939, 8050, 17916, 16807, 1, 28, 504, 5440, 35210, 135156, 286786, 262144, 1, 36, 882, 13387, 125730, 736401, 2642122, 5368728, 4782969, 1, 45, 1452, 29854, 388190, 3239491, 17424610, 58925728, 115089813, 100000000
Offset: 0

Views

Author

Gus Wiseman, Nov 28 2018

Keywords

Examples

			Triangle begins:
  1
  1     1
  1     3     3
  1     6    16    16
  1    10    51   127   125
  1    15   126   574  1347  1296
  1    21   266  1939  8050 17916 16807

Links

Crossrefs

Row sums are A322152. Last column is A000272.
Cf. A007718, A191646, A191970, A275421, A321155, A322114, A322115, A322137, A322147.

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[If[n==0,1,Length[Select[multsubs[multsubs[Range[k],2],n],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,5},{k,1,n+1}]
  • 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}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, 1/(1 - x + O(x*x^n) )^binomial(j+1, 2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Extensions

Offset corrected and terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

A322140 Number of labeled 2-connected multigraphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 1, 2, 7, 37, 262, 2312, 24338, 296928, 4112957, 63692909, 1089526922, 20389411551, 414146189901, 9070116944468, 212983762029683, 5336570227705763, 142083405456873290, 4004953714929148655, 119128974685786590410, 3728639072095285867881
Offset: 0

Views

Author

Gus Wiseman, Nov 27 2018

Keywords

Comments

We consider a single edge to be 2-connected, so a(1) = 1.

Links

Crossrefs

Cf. A002218, A002905, A013922, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

Programs

  • PARI
    seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, 1/(1 - y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

Extensions

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018
Showing 1-7 of 7 results.

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

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