A339065 -id:A339065 - cp's OEIS frontend

A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

1, 0, 1, 0, 2, 1, 1, 9, 9, 1, 15, 80, 90, 24, 1, 252, 1050, 1200, 450, 50, 1, 5005, 18018, 20475, 9100, 1575, 90, 1, 116280, 379848, 427329, 209475, 46550, 4410, 147, 1, 3108105, 9472320, 10548720, 5503680, 1433250, 183456, 10584, 224, 1

Offset: 0

Gus Wiseman, Jan 11 2024

			Triangle begins:
     1
     0     1
     0     2     1
     1     9     9     1
    15    80    90    24     1
   252  1050  1200   450    50     1
  5005 18018 20475  9100  1575    90     1
The loop-graphs counted in row n = 3 (loops shown as singletons):
  {12}{13}{23}  {1}{12}{13}  {1}{2}{12}  {1}{2}{3}
                {1}{12}{23}  {1}{2}{13}
                {1}{13}{23}  {1}{2}{23}
                {2}{12}{13}  {1}{3}{12}
                {2}{12}{23}  {1}{3}{13}
                {2}{13}{23}  {1}{3}{23}
                {3}{12}{13}  {2}{3}{12}
                {3}{12}{23}  {2}{3}{13}
                {3}{13}{23}  {2}{3}{23}

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Row sums are A014068, unlabeled version A000666.
Column k = 0 is A116508, covering version A367863.
The covering case is A368597.
The unlabeled version is A368836.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
A322661 counts labeled covering loop-graphs, connected A062740.
Cf. A057500, A079491, A339065, A368596, A368927.

Table[Length[Select[Subsets[Subsets[Range[n], {1,2}],{n}],Count[#,{_}]==k&]],{n,0,5},{k,0,n}]
T[n_,k_]:= Binomial[n,k]*Binomial[Binomial[n,2],n-k]; Table[T[n,k],{n,0,8},{k,0,n}]// Flatten (* Stefano Spezia, Jan 14 2024 *)

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k) \\ Andrew Howroyd, Jan 14 2024

T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k).

A339045 Number of connected loopless multigraphs with n edges rooted at two noninterchangeable vertices whose removal leaves a connected graph.

Original entry on oeis.org

1, 1, 4, 16, 69, 307, 1433, 6903, 34337, 175457, 919525, 4931233, 27023894, 151142376, 861880778, 5006906170, 29611120248, 178175786593, 1090266839041, 6781364484106, 42858210422338, 275127506187149, 1793418517202096, 11867326044069470, 79695273536227647

Offset: 1

Andrew Howroyd, Nov 25 2020

nonn

Cf. A338999, A339036, A339037, A339065.

\\ See A339065 for G.
InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoef(p,n)), vector(#v,n,1/n))}
seq(n)={my(A=O(x*x^n), g=G(2*n, x+A,[]), gr=G(2*n, x+A,[1])/g); InvEulerT(Vec(-1+G(2*n, x+A, [1,1])/(g*gr^2)))}

1/(Product_{k>=1} (1 - x^k)^a(k)) = f(x)/g(x)^2 where x*f(x) is the g.f. of A339037 and g(x) is the g.f. of A339036.

A339066 Number of unlabeled loopless multigraphs with n edges rooted at two indistinguishable vertices.

Original entry on oeis.org

1, 3, 12, 44, 171, 664, 2688, 11133, 47682, 210275, 955940, 4473128, 21532160, 106504216, 540824997, 2816636171, 15031261538, 82123830645, 458979942506, 2621982351176, 15298840540234, 91112889589166, 553492059017778, 3427579611162937, 21625096669854023, 138927108066308515

Offset: 0

Andrew Howroyd, Nov 22 2020

nonn

			The a(1) = 3 cases correspond to a single edge which can be attached to zero, one or both of the roots.

Cf. A050535, A007717 (one root), A339043, A339064, A339065.

seq[n_] := G[2n, x + O[x]^n, {1, 1}] + G[2n, x + O[x]^n, {2}] // CoefficientList[#/2, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, after Andrew Howroyd's code for G in A339065 *)

\\ See A339065 for G.
seq(n)={my(A=O(x*x^n)); Vec((G(2*n, x+A, [1, 1]) + G(2*n, x+A, [2]))/2)}

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A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

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A339045 Number of connected loopless multigraphs with n edges rooted at two noninterchangeable vertices whose removal leaves a connected graph.

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A339066 Number of unlabeled loopless multigraphs with n edges rooted at two indistinguishable vertices.

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