A005195 -id:A005195 - cp's OEIS frontend

A372194 Number of unlabeled graphs with n vertices and a unique triangle.

0, 0, 0, 1, 2, 7, 23, 102, 526, 3624, 32240, 382095, 5986945

Offset: 0

Gus Wiseman, Apr 24 2024

The labeled version is A372172.

			Representatives of the a(3) = 1 through a(6) = 23 graphs:
    12,13,23    12,13,23       12,13,23             12,13,23
                14,23,24,34    12,34,35,45          12,34,35,45
                               14,23,24,34          14,23,24,34
                               12,25,34,35,45       12,25,34,35,45
                               14,25,34,35,45       12,36,45,46,56
                               15,25,34,35,45       13,23,45,46,56
                               12,14,25,34,35,45    14,25,34,35,45
                                                    15,25,34,35,45
                                                    12,14,25,34,35,45
                                                    12,23,36,45,46,56
                                                    13,23,36,45,46,56
                                                    13,25,36,45,46,56
                                                    13,26,36,45,46,56
                                                    14,25,36,45,46,56
                                                    15,26,36,45,46,56
                                                    16,26,36,45,46,56
                                                    12,13,25,36,45,46,56
                                                    12,13,26,36,45,46,56
                                                    13,23,25,36,45,46,56
                                                    14,23,25,36,45,46,56
                                                    16,23,25,36,45,46,56
                                                    13,14,23,25,36,45,46,56
                                                    13,15,23,25,36,45,46,56

Gus Wiseman, The a(3) = 1 through a(6) = 23 graphs with a unique triangle.

For no triangles we have A006785, covering A372169.
Column k = 1 of A263340, covering A372173.
The labeled version is A372172.
The covering case is A372174, labeled A372171.
For all cycles (not just triangles): A236570, A372193, A372191, A372195.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002494 counts unlabeled covering graphs, labeled A006129.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A137917, A137918, A140637, A144958, A213434, A372168, A372170.

geng $n | countg -T1 # Georg Grasegger, Aug 03 2024

First differences are A372174.

a(11)-a(12) added by Georg Grasegger, Aug 03 2024

A095133 Triangle of numbers of forests on n nodes containing k trees.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 3, 2, 1, 1, 6, 6, 4, 2, 1, 1, 11, 11, 7, 4, 2, 1, 1, 23, 23, 14, 8, 4, 2, 1, 1, 47, 46, 29, 15, 8, 4, 2, 1, 1, 106, 99, 60, 32, 16, 8, 4, 2, 1, 1, 235, 216, 128, 66, 33, 16, 8, 4, 2, 1, 1, 551, 488, 284, 143, 69, 34, 16, 8, 4, 2, 1, 1, 1301, 1121, 636, 315, 149, 70, 34, 16, 8, 4, 2, 1, 1

Offset: 1

Eric W. Weisstein, May 29 2004

Row sums are A005195.

For k > n/2, T(n,k) = T(n-1,k-1). - Geoffrey Critzer, Oct 13 2012

			Triangle begins:
    1;
    1,  1;
    1,  1,  1;
    2,  2,  1,  1;
    3,  3,  2,  1,  1;
    6,  6,  4,  2,  1, 1;
   11, 11,  7,  4,  2, 1, 1;
   23, 23, 14,  8,  4, 2, 1, 1;
   47, 46, 29, 15,  8, 4, 2, 1, 1;
  106, 99, 60, 32, 16, 8, 4, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Peter Steinbach, Field Guide to Simple Graphs, Volume 3, Part 12 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 6 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)
Eric Weisstein's World of Mathematics, Forest

Cf. A005195 (row sums), A005196, A106240, A000055 (first column), A274937 (2nd column), A105821.
Limiting sequence of reversed rows gives A215930.
Reflected table is A136605. - Alois P. Heinz, Apr 11 2014

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*b(d), d=divisors(j)) *b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1,
      b(n)-(add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= (n, k)-> g(n, n, k):
seq(seq(a(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Aug 20 2012

nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);ft=Table[a[i]-Sum[a[j]a[i-j],{j,1,i/2}]+If[OddQ[i],0,a[i/2](a[i/2]+1)/2],{i,1,nn}];CoefficientList[Series[Product[1/(1-y x^i)^ft[[i]],{i,1,nn}],{x,0,20}],{x,y}]//Grid (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000055 *)

T(n, k) = sum over the partitions of n, 1M1 + 2M2 + ... + nMn, with exactly k parts, of Product_{i=1..n} binomial(A000055(i) + Mi - 1, Mi). - Washington Bomfim, May 12 2005

More terms from Vladeta Jovovic, Jun 03 2004

A165628 Number of 7-regular graphs (septic graphs) on 2n vertices.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 1547, 21609301, 733351105935, 42700033549946255, 4073194598236125134140, 613969628444792223023625238, 141515621596238755267618266465449

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-8)-regular graphs on 2n vertices.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Septic Graph

7-regular simple graphs: A014377 (connected), A165877 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), this sequence (k=7), A180260 (k=8).

A014377 = Cases[Import["https://oeis.org/A014377/b014377.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014377] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014377.

Cross-references edited by Jason Kimberley, Nov 07 2009 and Oct 17 2011
a(9)-a(11) from Andrew Howroyd, Mar 09 2020
a(12) from Andrew Howroyd, May 19 2020

A185335 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 9, 49, 455, 5784, 90940, 1620491, 31478651, 656784488, 14621878339, 345975756388

Offset: 0

Jason Kimberley, Jan 28 2011

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

3-regular simple graphs with girth at least 5: A014372 (connected), A185235 (disconnected), this sequence (not necessarily connected).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), A185334 (g=4), this sequence (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).
Not necessarily connected k-regular simple graphs with girth at least 5: A185315 (any k), A185305 (triangle); specified degree k: A185325 (k=2), this sequence (k=3).

A014372 = Cases[Import["https://oeis.org/A014372/b014372.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014372] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

This sequence is the Euler transformation of A014372.

A185344 Number of not necessarily connected 4-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 2, 12, 31, 220, 1606, 16829, 193900, 2452820, 32670332, 456028489, 6636066134, 100135577994, 1582718914660

Offset: 0

Jason Kimberley, Nov 03 2011

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

4-regular simple graphs with girth at least 4: A033886 (connected), A185244 (disconnected), this sequence (not necessarily connected).

Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: A008484 (k=2), A185334 (k=3), this sequence (k=4), A185354 (k=5), A185364 (k=6).

A033886 = Cases[Import["https://oeis.org/A033886/b033886.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A033886] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

This sequence is the Euler transformation of A033886.

a(n) = A033886(n) + A185244(n).

A185354 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 7, 388, 406824, 1125022326, 3813549359275

Offset: 0

Jason Kimberley, Nov 04 2011

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

5-regular simple graphs on 2n vertices with girth at least 4: A058275 (connected), A185254 (disconnected), this sequence (not necessarily connected).

Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: A008484 (k=2), A185334 (k=3), A185344 (k=4), this sequence (k=5), A185364 (k=6).

A058275 = Cases[Import["https://oeis.org/A058275/b058275.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A058275]  (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

This sequence is the Euler transformation of A058275.

a(n) = A058275(n) + A185254(n).

A185364 Not necessarily connected 6-regular simple graphs on n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 9, 6, 267, 3727, 483012, 69823723, 14836130862

Offset: 0

Jason Kimberley, Dec 07 2011

First differs from A058276 at n=24.

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

6-regular simple graphs with girth at least 4: A058276 (connected), A185264 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: A008484 (k=2), A185334 (k=3), A185344 (k=4), A185354 (k=5), this sequence (k=6).
Cf. A184964.

A058276 = Cases[Import["https://oeis.org/A058276/b058276.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A058276] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

This sequence is the Euler transformation of A058276.

a(n) = A058276(n) + A185264(n).

A058338 Number of digraphs with indegree = outdegree at each vertex, or Eulerian digraphs (including disconnected graphs) with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 18, 111, 2292, 181519, 51294836, 48814342597, 157166848652408, 1736132851915020181, 66689407510956897981608, 9012860249531358132125181323, 4328664025050045497712238253179872, 7451990930474005836163802713084971814275, 46329549163592383403451764167315165130121820112

Offset: 0

N. J. A. Sloane

Every regular tournament (A096368) is a Eulerian digraph. Similar methods may be used to compute terms of this sequence. - Andrew Howroyd, Apr 12 2020

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 219 (but there is an error).
Ronald C. Read, email to N. J. A. Sloane, 28 August, 2000.

Cf. A007080 (labeled), A058337 (connected), A096368, A308161, A308111.

A058337 = Cases[Import["https://oeis.org/A058337/b058337.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A058337] (* Jean-François Alcover, Feb 23 2020, upated Mar 18 2020 *)

Euler transform of A058337.

a(7) added using A058337 by Falk Hüffner, Dec 03 2015
a(8) and a(9) added using A058337 by Brendan McKay, May 05 2019
Terms a(10) and beyond from Andrew Howroyd, Apr 12 2020

A180260 Number of not necessarily connected 8-regular simple graphs on n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 6, 94, 10786, 3459386, 1470293676, 733351105935, 423187422492342, 281341168330848874, 214755319657939505396, 187549729101764460261505, 186685399408147545744203915, 210977245260028917322933165888

Offset: 0

Jason Kimberley, Jan 17 2011

The Euler transformation currently does nothing: for n < 18, a(n) = A014378(n).

			The a(0)=1 graph is K_0 (vacuously 8-regular).
The a(9)=1 graph is K_9.

Eric Weisstein's World of Mathematics, Octic Graph

8-regular simple graphs: A014378 (connected), A165878 (disconnected), this sequence (not necessarily connected).
Not necessarily connected regular simple graphs: A005176 (any degree), A051031 (triangular array), specified degree k: A000012 (k=0), A000012 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), A165627 (k=6), A165628 (k=7), this sequence (k=8).
8-regular not necessarily connected graphs: this sequence (simple graphs), A129437 (multigraphs with loops allowed), A129426 (multigraphs with loops forbidden).

A014378 = Cases[Import["https://oeis.org/A014378/b014378.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014378] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014378.

a(17)-a(22) from Andrew Howroyd, Mar 08 2020

A005196 a(n) = Sum_t t*F(n,t), where F(n,t) (see A095133) is the number of forests with n (unlabeled) nodes and exactly t trees.

Original entry on oeis.org

1, 3, 6, 13, 24, 49, 93, 190, 381, 803, 1703, 3755, 8401, 19338, 45275, 108229, 262604, 647083, 1613941, 4072198, 10374138, 26663390, 69056163, 180098668, 472604314, 1247159936, 3307845730, 8814122981, 23585720703, 63359160443, 170815541708, 462049250165

Offset: 1

N. J. A. Sloane

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 = 1..300
E. M. Palmer and A. J. Schwenk, On the number of trees in a random forest, J. Combin. Theory, B 27 (1979), 109-121. [The 17th entry is wrong]
Eric Weisstein's World of Mathematics, Forest

Cf. A000055, A005195, A095133.

with(numtheory):
b:= proc(n) option remember; local d, j; `if` (n<=1, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
t:= proc(n) option remember; local k; `if` (n=0, 1,
      b(n)-(add(b(k)*b(n-k), k=0..n)-`if`(irem(n, 2)=0, b(n/2), 0))/2)
    end:
g:= proc(n, i, p) option remember; `if`(p>n, 0, `if`(n=0, 1,
      `if`(min(i, p)<1, 0, add(g(n-i*j, i-1, p-j) *
       binomial(t(i)+j-1, j), j=0..min(n/i, p)))))
    end:
a:= n-> add(k*g(n, n, k), k=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 20 2012

nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);ft=Table[a[i]-Sum[a[j]a[i-j],{j,1,i/2}]+If[OddQ[i],0,a[i/2](a[i/2]+1)/2],{i,1,nn}];CoefficientList[Series[D[Product[1/(1-y x^i)^ft[[i]],{i,1,nn}],y]/.y->1,{x,0,20}],x]  (* Geoffrey Critzer, Oct 13 2012, after code given by Robert A. Russell in A000055 *)

To get a(n), take row n of the triangle in A095133, multiply successive terms by 1, 2, 3, ... and sum. E.g., a(4) = 1*2 + 2*2 + 3*1 + 4*1 = 13.

More terms from Vladeta Jovovic, Jun 03 2004

Definition clarified by N. J. A. Sloane, May 29 2012

cp's OEIS Frontend

A372194 Number of unlabeled graphs with n vertices and a unique triangle.

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nauty

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A095133 Triangle of numbers of forests on n nodes containing k trees.

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Maple

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A165628 Number of 7-regular graphs (septic graphs) on 2n vertices.

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Mathematica

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A185335 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 5.

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Mathematica

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A185344 Number of not necessarily connected 4-regular simple graphs on n vertices with girth at least 4.

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Mathematica

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A185354 Number of not necessarily connected 5-regular simple graphs on 2n vertices with girth at least 4.

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Programs

Mathematica

Formula

A185364 Not necessarily connected 6-regular simple graphs on n vertices with girth at least 4.

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Mathematica

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A058338 Number of digraphs with indegree = outdegree at each vertex, or Eulerian digraphs (including disconnected graphs) with n nodes.

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