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A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.

%I A369199 #11 Feb 02 2024 19:40:20
%S A369199 1,0,1,0,1,3,1,0,0,6,17,15,6,1,0,0,3,46,150,228,206,120,45,10,1,0,0,0,
%T A369199 45,465,1803,3965,5835,6210,4955,2998,1365,455,105,15,1,0,0,0,15,645,
%U A369199 5991,27364,79470,165555,264050,334713,344526,291200,202860,116190,54258,20349,5985,1330,210,21,1
%N A369199 Irregular triangle read by rows where T(n,k) is the number of labeled loop-graphs covering n vertices with k edges.
%H A369199 Andrew Howroyd, <a href="/A369199/b369199.txt">Table of n, a(n) for n = 0..1560</a> (rows 0..20)
%F A369199 E.g.f.: exp(-x) * (Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - _Andrew Howroyd_, Feb 02 2024
%e A369199 Triangle begins:
%e A369199    1
%e A369199    0   1
%e A369199    0   1   3   1
%e A369199    0   0   6  17  15   6   1
%e A369199    0   0   3  46 150 228 206 120  45  10   1
%e A369199 Row n = 3 counts the following loop-graphs (loops shown as singletons):
%e A369199   {1,23}   {1,2,3}     {1,2,3,12}    {1,2,3,12,13}   {1,2,3,12,13,23}
%e A369199   {2,13}   {1,2,13}    {1,2,3,13}    {1,2,3,12,23}
%e A369199   {3,12}   {1,2,23}    {1,2,3,23}    {1,2,3,13,23}
%e A369199   {12,13}  {1,3,12}    {1,2,12,13}   {1,2,12,13,23}
%e A369199   {12,23}  {1,3,23}    {1,2,12,23}   {1,3,12,13,23}
%e A369199   {13,23}  {1,12,13}   {1,2,13,23}   {2,3,12,13,23}
%e A369199            {1,12,23}   {1,3,12,13}
%e A369199            {1,13,23}   {1,3,12,23}
%e A369199            {2,3,12}    {1,3,13,23}
%e A369199            {2,3,13}    {1,12,13,23}
%e A369199            {2,12,13}   {2,3,12,13}
%e A369199            {2,12,23}   {2,3,12,23}
%e A369199            {2,13,23}   {2,3,13,23}
%e A369199            {3,12,13}   {2,12,13,23}
%e A369199            {3,12,23}   {3,12,13,23}
%e A369199            {3,13,23}
%e A369199            {12,13,23}
%t A369199 Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}],Length[Union@@#]==n&]],{n,0,5},{k,0,Binomial[n+1,2]}]
%o A369199 (PARI) T(n)={[Vecrev(p) | p<-Vec(serlaplace(exp(-x + O(x*x^n))*(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!)))) ]}
%o A369199 { my(A=T(6)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Feb 02 2024
%Y A369199 The version without loops is A054548.
%Y A369199 This is the covering case of A084546.
%Y A369199 Column sums are A173219.
%Y A369199 Row sums are A322661, unlabeled A322700.
%Y A369199 The connected case is A369195, without loops A062734.
%Y A369199 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A369199 A006125 counts simple graphs; also loop-graphs if shifted left.
%Y A369199 A006129 counts covering graphs, unlabeled A002494.
%Y A369199 A368927 counts choosable loop-graphs, covering A369140.
%Y A369199 A369141 counts non-choosable loop-graphs, covering A369142.
%Y A369199 Cf. A000666, A006649, A062740, A066383, A133686, A368597, A369196, A369197.
%K A369199 nonn,tabf
%O A369199 0,6
%A A369199 _Gus Wiseman_, Jan 18 2024