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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.

%I A322147 #12 Apr 15 2021 21:39:24
%S A322147 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 A322147 2565,11436,16807,0,0,0,116,4615,47100,185944,262144,0,0,0,45,5540,
%U A322147 121185,987567,3533720,4782969,0,0,0,10,4720,220075,3376450,23315936,76826061,100000000
%N 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.
%H A322147 Andrew Howroyd, <a href="/A322147/b322147.txt">Table of n, a(n) for n = 0..1274</a>
%e A322147 Triangle begins:
%e A322147   1
%e A322147   1     1
%e A322147   0     2     3
%e A322147   0     1    10    16
%e A322147   0     0    12    79   125
%e A322147   0     0     6   162   847  1296
%e A322147   0     0     1   179  2565 11436 16807
%t A322147 multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t A322147 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 A322147 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 A322147 (PARI)
%o A322147 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 A322147 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 A322147 { my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ _Andrew Howroyd_, Nov 29 2018
%Y A322147 Row sums are A322151. Last column is A000272.
%Y A322147 Column sums are A062740.
%Y A322147 Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.
%K A322147 nonn,tabl
%O A322147 0,5
%A A322147 _Gus Wiseman_, Nov 28 2018
%E A322147 Terms a(28) and beyond from _Andrew Howroyd_, Nov 29 2018