cp's OEIS Frontend

A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

%I A322148 #8 Nov 29 2018 15:10:51
%S A322148 1,1,1,1,3,3,1,6,16,16,1,10,51,127,125,1,15,126,574,1347,1296,1,21,
%T A322148 266,1939,8050,17916,16807,1,28,504,5440,35210,135156,286786,262144,1,
%U A322148 36,882,13387,125730,736401,2642122,5368728,4782969,1,45,1452,29854,388190,3239491,17424610,58925728,115089813,100000000
%N A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.
%H A322148 Andrew Howroyd, <a href="/A322148/b322148.txt">Table of n, a(n) for n = 0..1274</a>
%e A322148 Triangle begins:
%e A322148   1
%e A322148   1     1
%e A322148   1     3     3
%e A322148   1     6    16    16
%e A322148   1    10    51   127   125
%e A322148   1    15   126   574  1347  1296
%e A322148   1    21   266  1939  8050 17916 16807
%t A322148 multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
%t A322148 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 A322148 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 A322148 (PARI)
%o A322148 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 A322148 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 A322148 { my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ _Andrew Howroyd_, Nov 29 2018
%Y A322148 Row sums are A322152. Last column is A000272.
%Y A322148 Cf. A007718, A191646, A191970, A275421, A321155, A322114, A322115, A322137, A322147.
%K A322148 nonn,tabl
%O A322148 0,5
%A A322148 _Gus Wiseman_, Nov 28 2018
%E A322148 Offset corrected and terms a(28) and beyond from _Andrew Howroyd_, Nov 29 2018