A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

1, 1, 2, 10, 79, 847, 11436, 185944, 3533720, 76826061, 1880107840, 51139278646, 1530376944768, 49965900317755, 1767387701671424, 67325805434672100, 2747849045156064256, 119626103584870552921, 5533218319763109888000, 270982462739224265922466

Offset: 0

Author

Andrew Howroyd, Jan 10 2024

Keywords

• nonn

Comments

Exponential transform appears to be A333331. - Gus Wiseman, Feb 12 2024

Examples

			From _Gus Wiseman_, Feb 12 2024: (Start)
The a(0) = 1 through a(3) = 10 loop-graphs:
  {}  {11}  {11,12}  {11,12,13}
            {22,12}  {11,12,23}
                     {11,13,23}
                     {22,12,13}
                     {22,12,23}
                     {22,13,23}
                     {33,12,13}
                     {33,12,23}
                     {33,13,23}
                     {12,13,23}
(End)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Graph Loop.

Crossrefs

Cf. A000169, A333331, A063170, A053506.

This is the connected covering case of A014068.

The case without loops is A057500, covering case of A370317.

Allowing any number of edges gives A062740, connected case of A322661.

This is the connected case of A368597.

The unlabeled version is A368983, connected case of A368984.

For at most n edges we have A369197.

A000085 counts set partitions into singletons or pairs.

A006129 counts covering graphs, connected A001187.

Cf. A000272, A000666, A054780, A116508, A136556, A367863, A368600.

Programs

Maple

egf:= (L-> 1-L/2-log(1+L)/2-L^2/4)(LambertW(-x)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 10 2024

PARI

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(-log(1-t)/2 + t/2 - t^2/4 + 1))}

Formula

a(n) = A000169(n) + A057500(n) for n > 0.

E.g.f.: 1 - log(1-T(x))/2 + T(x)/2 - T(x)^2/4 where T(x) = -LambertW(-x) is the e.g.f. of A000169.

From Peter Luschny, Jan 10 2024: (Start)

a(n) = (exp(n)*Gamma(n + 1, n) - (n - 1)*n^(n - 1))/(2*n) for n > 0.

a(n) = (1/2)*(A063170(n)/n - A053506(n)) for n > 0. (End)