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A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.

%I A368928 #18 Jan 14 2024 16:11:04
%S A368928 1,0,1,0,2,1,1,9,9,1,15,80,90,24,1,252,1050,1200,450,50,1,5005,18018,
%T A368928 20475,9100,1575,90,1,116280,379848,427329,209475,46550,4410,147,1,
%U A368928 3108105,9472320,10548720,5503680,1433250,183456,10584,224,1
%N A368928 Triangle read by rows where T(n,k) is the number of labeled loop-graphs with n vertices and n edges, k of which are loops.
%H A368928 Andrew Howroyd, <a href="/A368928/b368928.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A368928 T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k).
%e A368928 Triangle begins:
%e A368928      1
%e A368928      0     1
%e A368928      0     2     1
%e A368928      1     9     9     1
%e A368928     15    80    90    24     1
%e A368928    252  1050  1200   450    50     1
%e A368928   5005 18018 20475  9100  1575    90     1
%e A368928 The loop-graphs counted in row n = 3 (loops shown as singletons):
%e A368928   {12}{13}{23}  {1}{12}{13}  {1}{2}{12}  {1}{2}{3}
%e A368928                 {1}{12}{23}  {1}{2}{13}
%e A368928                 {1}{13}{23}  {1}{2}{23}
%e A368928                 {2}{12}{13}  {1}{3}{12}
%e A368928                 {2}{12}{23}  {1}{3}{13}
%e A368928                 {2}{13}{23}  {1}{3}{23}
%e A368928                 {3}{12}{13}  {2}{3}{12}
%e A368928                 {3}{12}{23}  {2}{3}{13}
%e A368928                 {3}{13}{23}  {2}{3}{23}
%t A368928 Table[Length[Select[Subsets[Subsets[Range[n], {1,2}],{n}],Count[#,{_}]==k&]],{n,0,5},{k,0,n}]
%t A368928 T[n_,k_]:= Binomial[n,k]*Binomial[Binomial[n,2],n-k]; Table[T[n,k],{n,0,8},{k,0,n}]// Flatten (* _Stefano Spezia_, Jan 14 2024 *)
%o A368928 (PARI) T(n,k) = binomial(n,k)*binomial(binomial(n,2),n-k) \\ _Andrew Howroyd_, Jan 14 2024
%Y A368928 Row sums are A014068, unlabeled version A000666.
%Y A368928 Column k = 0 is A116508, covering version A367863.
%Y A368928 The covering case is A368597.
%Y A368928 The unlabeled version is A368836.
%Y A368928 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A368928 A006125 counts graphs, unlabeled A000088.
%Y A368928 A006129 counts covering graphs, unlabeled A002494.
%Y A368928 A058891 counts set-systems (without singletons A016031), unlabeled A000612.
%Y A368928 A322661 counts labeled covering loop-graphs, connected A062740.
%Y A368928 Cf. A057500, A079491, A339065, A368596, A368927.
%K A368928 nonn,tabl
%O A368928 0,5
%A A368928 _Gus Wiseman_, Jan 11 2024