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.

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

Author

Gus Wiseman, Nov 28 2018

Keywords

• nonn
• tabl

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

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

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