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Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.
3

%I #8 Nov 29 2018 15:10:51

%S 1,1,1,1,3,3,1,6,16,16,1,10,51,127,125,1,15,126,574,1347,1296,1,21,

%T 266,1939,8050,17916,16807,1,28,504,5440,35210,135156,286786,262144,1,

%U 36,882,13387,125730,736401,2642122,5368728,4782969,1,45,1452,29854,388190,3239491,17424610,58925728,115089813,100000000

%N Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

%H Andrew Howroyd, <a href="/A322148/b322148.txt">Table of n, a(n) for n = 0..1274</a>

%e Triangle begins:

%e 1

%e 1 1

%e 1 3 3

%e 1 6 16 16

%e 1 10 51 127 125

%e 1 15 126 574 1347 1296

%e 1 21 266 1939 8050 17916 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[multsubs[multsubs[Range[k],2],n],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,5},{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/(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 A322152. Last column is A000272.

%Y Cf. A007718, A191646, A191970, A275421, A321155, A322114, A322115, A322137, A322147.

%K nonn,tabl

%O 0,5

%A _Gus Wiseman_, Nov 28 2018

%E Offset corrected and terms a(28) and beyond from _Andrew Howroyd_, Nov 29 2018