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%I #14 Jan 14 2024 15:48:00
%S 1,0,1,0,1,1,1,2,1,1,2,5,3,1,1,5,12,7,3,1,1,14,29,19,8,3,1,1,35,75,47,
%T 21,8,3,1,1,97,191,127,54,22,8,3,1,1,264,504,331,149,56,22,8,3,1,1,
%U 733,1339,895,395,156,57,22,8,3,1,1
%N Triangle read by rows where T(n,k) is the number of unlabeled loop-graphs on n vertices with k loops and n-k non-loops such that it is possible to choose a different element from each edge.
%C Also the number of unlabeled loop-graphs covering n vertices with k loops and n-k non-loops such that each connected component has the same number of edges as vertices.
%H Andrew Howroyd, <a href="/A368926/b368926.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 1 2 1 1
%e 2 5 3 1 1
%e 5 12 7 3 1 1
%e 14 29 19 8 3 1 1
%e 35 75 47 21 8 3 1 1
%t Table[Length[Union[sysnorm /@ Select[Subsets[Subsets[Range[n],{1,2}],{n}],Count[#,{_}]==k && Length[Select[Tuples[#],UnsameQ@@#&]]!=0&]]], {n,0,5},{k,0,n}]
%o (PARI) \\ TreeGf gives gf of A000081; G(n,1) is gf of A368983.
%o TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
%o G(n,y)={my(t=TreeGf(n)); my(g(e)=subst(t + O(x*x^(n\e)), x, x^e) + O(x*x^n)); 1 + (sum(d=1, n, eulerphi(d)/d*log(1/(1-g(d)))) + ((1+g(1))^2/(1-g(2))-1)/2 - (g(1)^2 + g(2)))/2 + (y-1)*g(1)}
%o EulerMTS(p)={my(n=serprec(p,x)-1,vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i))}
%o T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n,y) - 1))]}
%o { my(A=T(8)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 14 2024
%Y The case of a unique choice is A106234, row sums A000081.
%Y Column k = 0 is A137917, labeled version A137916.
%Y Without the choice condition we have A368836.
%Y The labeled version is A368924, row sums maybe A333331.
%Y Row sums are A368984, complement A368835.
%Y A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A006125 counts graphs, unlabeled A000088.
%Y A006129 counts covering graphs, unlabeled A002494.
%Y A014068 counts loop-graphs, unlabeled A000666.
%Y A322661 counts labeled covering half-loop-graphs, connected A062740.
%Y Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.
%K nonn,tabl
%O 0,8
%A _Gus Wiseman_, Jan 13 2024
%E a(36) onwards from _Andrew Howroyd_, Jan 14 2024
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