A322152 - OEIS

%I #5 Nov 28 2018 20:57:30

%S 1,2,7,39,314,3359,45000,725269,13670256,295099184,7179749707,

%T 194399095705,5797793490859,188855813757729,6671188010874785,

%U 254007814638737649,10370334196814589256,451923738493729293016,20937747226064522726151,1027666505638118490940059

%N Number of labeled connected multigraphs with loops with n edges (the vertices are {1,2,...,k} for some k).

%H Andrew Howroyd, <a href="/A322152/b322152.txt">Table of n, a(n) for n = 0..200</a>

%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[Length[Select[multsubs[multsubs[Range[n+1],2],n],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]

%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 seq(n)={Vec(vecsum(Connected(vector(2*n, j, 1/(1 - x + O(x*x^n))^binomial(j+1,2)))))} \\ _Andrew Howroyd_, Nov 28 2018

%Y Row sums of A322148. The unlabeled version is A007719.

%Y Cf. A000272, A000664, A007718, A191646, A191970, A322114, A322115, A322147, A322151.

%K nonn

%O 0,2

%A _Gus Wiseman_, Nov 28 2018

%E Terms a(7) and beyond from _Andrew Howroyd_, Nov 28 2018