A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

1, 0, 1, 0, 1, 3, 1, 0, 0, 6, 17, 15, 6, 1, 0, 0, 3, 46, 150, 228, 206, 120, 45, 10, 1, 0, 0, 0, 45, 465, 1803, 3965, 5835, 6210, 4955, 2998, 1365, 455, 105, 15, 1, 0, 0, 0, 15, 645, 5991, 27364, 79470, 165555, 264050, 334713, 344526, 291200, 202860, 116190, 54258, 20349, 5985, 1330, 210, 21, 1

Offset: 0

Author

Gus Wiseman, Jan 18 2024

Keywords

• nonn
• tabf

Examples

			Triangle begins:
   1
   0   1
   0   1   3   1
   0   0   6  17  15   6   1
   0   0   3  46 150 228 206 120  45  10   1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  {1,23}   {1,2,3}     {1,2,3,12}    {1,2,3,12,13}   {1,2,3,12,13,23}
  {2,13}   {1,2,13}    {1,2,3,13}    {1,2,3,12,23}
  {3,12}   {1,2,23}    {1,2,3,23}    {1,2,3,13,23}
  {12,13}  {1,3,12}    {1,2,12,13}   {1,2,12,13,23}
  {12,23}  {1,3,23}    {1,2,12,23}   {1,3,12,13,23}
  {13,23}  {1,12,13}   {1,2,13,23}   {2,3,12,13,23}
           {1,12,23}   {1,3,12,13}
           {1,13,23}   {1,3,12,23}
           {2,3,12}    {1,3,13,23}
           {2,3,13}    {1,12,13,23}
           {2,12,13}   {2,3,12,13}
           {2,12,23}   {2,3,12,23}
           {2,13,23}   {2,3,13,23}
           {3,12,13}   {2,12,13,23}
           {3,12,23}   {3,12,13,23}
           {3,13,23}
           {12,13,23}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

Crossrefs

The version without loops is A054548.

This is the covering case of A084546.

Column sums are A173219.

Row sums are A322661, unlabeled A322700.

The connected case is A369195, without loops A062734.

A000085, A100861, A111924 count set partitions into singletons or pairs.

A006125 counts simple graphs; also loop-graphs if shifted left.

A006129 counts covering graphs, unlabeled A002494.

A368927 counts choosable loop-graphs, covering A369140.

A369141 counts non-choosable loop-graphs, covering A369142.

Cf. A000666, A006649, A062740, A066383, A133686, A368597, A369196, A369197.

Programs

Mathematica

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}],Length[Union@@#]==n&]],{n,0,5},{k,0,Binomial[n+1,2]}]

PARI

T(n)={[Vecrev(p) | p<-Vec(serlaplace(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

Formula

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