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A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

%I A369195 #20 Feb 02 2024 23:56:15
%S A369195 1,0,1,0,1,2,1,0,0,3,10,12,6,1,0,0,0,16,79,162,179,116,45,10,1,0,0,0,
%T A369195 0,125,847,2565,4615,5540,4720,2948,1360,455,105,15,1,0,0,0,0,0,1296,
%U A369195 11436,47100,121185,220075,301818,325578,282835,200115,115560,54168,20343,5985,1330,210,21,1
%N A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.
%C A369195 This sequence excludes the graph consisting of a single isolated vertex without a loop. - _Andrew Howroyd_, Feb 02 2024
%H A369195 Andrew Howroyd, <a href="/A369195/b369195.txt">Table of n, a(n) for n = 0..1560</a> (rows 0..20)
%F A369195 E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - _Andrew Howroyd_, Feb 02 2024
%e A369195 Triangle begins:
%e A369195     1
%e A369195     0    1
%e A369195     0    1    2    1
%e A369195     0    0    3   10   12    6    1
%e A369195     0    0    0   16   79  162  179  116   45   10    1
%e A369195 Row n = 3 counts the following loop-graphs (loops shown as singletons):
%e A369195   .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
%e A369195         {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
%e A369195         {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
%e A369195                  {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
%e A369195                  {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
%e A369195                  {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
%e A369195                  {3,12,13}   {1,12,13,23}
%e A369195                  {3,12,23}   {2,3,12,13}
%e A369195                  {3,13,23}   {2,3,12,23}
%e A369195                  {12,13,23}  {2,3,13,23}
%e A369195                              {2,12,13,23}
%e A369195                              {3,12,13,23}
%t A369195 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A369195 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]
%o A369195 (PARI) T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
%o A369195 { my(A=T(6)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Feb 02 2024
%Y A369195 Row lengths are A000124.
%Y A369195 Diagonal T(n,n-1) is A000272, rooted A000169.
%Y A369195 The case without loops is A062734.
%Y A369195 Row sums are A062740.
%Y A369195 Transpose is A322147.
%Y A369195 Column sums are A322151.
%Y A369195 Diagonal T(n,n) is A368951, connected case of A368597.
%Y A369195 Connected case of A369199, without loops A054548.
%Y A369195 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A369195 A000666 counts unlabeled loop-graphs.
%Y A369195 A001187 counts connected graphs, unlabeled A001349.
%Y A369195 A006125 counts simple graphs, also loop-graphs if shifted left.
%Y A369195 A006129 counts covering graphs, unlabeled A002494.
%Y A369195 A322661 counts covering loop-graphs, unlabeled A322700.
%Y A369195 A368927 counts choosable loop-graphs, covering A369140.
%Y A369195 A369141 counts non-choosable loop-graphs, covering A369142.
%Y A369195 Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.
%K A369195 nonn,tabf
%O A369195 0,6
%A A369195 _Gus Wiseman_, Jan 19 2024