A368926 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.

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, 21, 8, 3, 1, 1, 97, 191, 127, 54, 22, 8, 3, 1, 1, 264, 504, 331, 149, 56, 22, 8, 3, 1, 1, 733, 1339, 895, 395, 156, 57, 22, 8, 3, 1, 1

Offset

0,8

Comments

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.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Example

Triangle begins:
   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 21  8  3  1  1

Mathematica

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}]

Prog

(PARI) \\ TreeGf gives gf of A000081; G(n, 1) is gf of A368983.
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)}
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)}
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))}
T(n)={[Vecrev(p) | p <- Vec(EulerMTS(G(n, y) - 1))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 14 2024

Crossrefs

The case of a unique choice is A106234, row sums A000081.

Column k = 0 is A137917, labeled version A137916.

Without the choice condition we have A368836.

The labeled version is A368924, row sums maybe A333331.

Row sums are A368984, complement A368835.

A000085, A100861, A111924 count set partitions into singletons or pairs.

A006125 counts graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A014068 counts loop-graphs, unlabeled A000666.

A322661 counts labeled covering half-loop-graphs, connected A062740.

Cf. A057500, A116508, A133686, A367863, A367869, A368596, A368597, A368598, A368601, A368836, A368927.

Sequence in context: A305313 A159046 A029937 * A289772 A283615 A172483

Adjacent sequences: A368923 A368924 A368925 * A368927 A368928 A368929

Keyword

nonn,tabl

Author

Gus Wiseman, Jan 13 2024

Extensions

a(36) onwards from Andrew Howroyd, Jan 14 2024

Status

approved