A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000

Offset: 0

Author

Gus Wiseman, Jan 17 2024

Keywords

• nonn

Examples

			The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
  .  {{1}}  {{1,2}}      {{1,2},{1,3}}
            {{1},{1,2}}  {{1,2},{2,3}}
            {{2},{1,2}}  {{1,3},{2,3}}
                         {{1},{1,2},{1,3}}
                         {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1,2},{1,3},{2,3}}
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

The minimal case is A000272.

Connected case of A066383 and A369196, loopless A369192 and A369193.

The loopless case is A129271, connected case of A369191.

The case of equality is A368951, connected case of A368597.

This is the connected case of A369194.

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

A001187 counts connected graphs, unlabeled A001349.

A006125 counts (simple) graphs, unlabeled A000088.

A006129 counts covering graphs, unlabeled A002494.

A054548 counts graphs covering n vertices with k edges, with loops A369199.

A062740 counts connected loop-graphs.

A322661 counts covering loop-graphs, unlabeled A322700.

A368927 counts choosable loop-graphs, covering A369140.

A369141 counts non-choosable loop-graphs, covering A369142.

Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.

Programs

PARI

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024

Formula

Logarithmic transform of A368927.

From Andrew Howroyd, Feb 02 2024: (Start)

a(n) = A000169(n) + A129271(n).

E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)

Extensions

a(0) changed to 1 and a(7) onwards from Andrew Howroyd, Feb 02 2024