cp's OEIS Frontend

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

%I A368951 #23 Feb 22 2024 09:04:41
%S A368951 1,1,2,10,79,847,11436,185944,3533720,76826061,1880107840,51139278646,
%T A368951 1530376944768,49965900317755,1767387701671424,67325805434672100,
%U A368951 2747849045156064256,119626103584870552921,5533218319763109888000,270982462739224265922466
%N A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.
%C A368951 Exponential transform appears to be A333331. - _Gus Wiseman_, Feb 12 2024
%H A368951 Andrew Howroyd, <a href="/A368951/b368951.txt">Table of n, a(n) for n = 0..200</a>
%H A368951 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GraphLoop.html">Graph Loop</a>.
%F A368951 a(n) = A000169(n) + A057500(n) for n > 0.
%F A368951 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.
%F A368951 From _Peter Luschny_, Jan 10 2024: (Start)
%F A368951 a(n) = (exp(n)*Gamma(n + 1, n) - (n - 1)*n^(n - 1))/(2*n) for n > 0.
%F A368951 a(n) = (1/2)*(A063170(n)/n - A053506(n)) for n > 0.  (End)
%e A368951 From _Gus Wiseman_, Feb 12 2024: (Start)
%e A368951 The a(0) = 1 through a(3) = 10 loop-graphs:
%e A368951   {}  {11}  {11,12}  {11,12,13}
%e A368951             {22,12}  {11,12,23}
%e A368951                      {11,13,23}
%e A368951                      {22,12,13}
%e A368951                      {22,12,23}
%e A368951                      {22,13,23}
%e A368951                      {33,12,13}
%e A368951                      {33,12,23}
%e A368951                      {33,13,23}
%e A368951                      {12,13,23}
%e A368951 (End)
%p A368951 egf:= (L-> 1-L/2-log(1+L)/2-L^2/4)(LambertW(-x)):
%p A368951 a:= n-> n!*coeff(series(egf, x, n+1), x, n):
%p A368951 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 10 2024
%o A368951 (PARI) seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(-log(1-t)/2 + t/2 - t^2/4 + 1))}
%Y A368951 Cf. A000169, A333331, A063170, A053506.
%Y A368951 This is the connected covering case of A014068.
%Y A368951 The case without loops is A057500, covering case of A370317.
%Y A368951 Allowing any number of edges gives A062740, connected case of A322661.
%Y A368951 This is the connected case of A368597.
%Y A368951 The unlabeled version is A368983, connected case of A368984.
%Y A368951 For at most n edges we have A369197.
%Y A368951 A000085 counts set partitions into singletons or pairs.
%Y A368951 A006129 counts covering graphs, connected A001187.
%Y A368951 Cf. A000272, A000666, A054780, A116508, A136556, A367863, A368600.
%K A368951 nonn
%O A368951 0,3
%A A368951 _Andrew Howroyd_, Jan 10 2024