%I #12 Apr 15 2021 21:39:24

%S 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,

%T 2565,11436,16807,0,0,0,116,4615,47100,185944,262144,0,0,0,45,5540,

%U 121185,987567,3533720,4782969,0,0,0,10,4720,220075,3376450,23315936,76826061,100000000

%N 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.

%H Andrew Howroyd, <a href="/A322147/b322147.txt">Table of n, a(n) for n = 0..1274</a>

%e Triangle begins:

%e 1

%e 1 1

%e 0 2 3

%e 0 1 10 16

%e 0 0 12 79 125

%e 0 0 6 162 847 1296

%e 0 0 1 179 2565 11436 16807

%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];

%t 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]]]]]]]]];

%t 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}]

%o (PARI)

%o 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}

%o 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]])}

%o { my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ _Andrew Howroyd_, Nov 29 2018

%Y Row sums are A322151. Last column is A000272.

%Y Column sums are A062740.

%Y Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Nov 28 2018

%E Terms a(28) and beyond from _Andrew Howroyd_, Nov 29 2018