cp's OEIS Frontend

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

%I A369197 #13 Feb 02 2024 16:10:28
%S A369197 1,1,3,13,95,972,12732,202751,3795864,81609030,1980107840,53497226337,
%T A369197 1592294308992,51758060711792,1824081614046720,69272000503031475,
%U A369197 2819906639193992192,122488526636380368714,5654657850859704139776,276462849597009068108405,14270030377126199463936000
%N A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.
%H A369197 Andrew Howroyd, <a href="/A369197/b369197.txt">Table of n, a(n) for n = 0..200</a>
%F A369197 Logarithmic transform of A368927.
%F A369197 From _Andrew Howroyd_, Feb 02 2024: (Start)
%F A369197 a(n) = A000169(n) + A129271(n).
%F A369197 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)
%e A369197 The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
%e A369197   .  {{1}}  {{1,2}}      {{1,2},{1,3}}
%e A369197             {{1},{1,2}}  {{1,2},{2,3}}
%e A369197             {{2},{1,2}}  {{1,3},{2,3}}
%e A369197                          {{1},{1,2},{1,3}}
%e A369197                          {{1},{1,2},{2,3}}
%e A369197                          {{1},{1,3},{2,3}}
%e A369197                          {{2},{1,2},{1,3}}
%e A369197                          {{2},{1,2},{2,3}}
%e A369197                          {{2},{1,3},{2,3}}
%e A369197                          {{3},{1,2},{1,3}}
%e A369197                          {{3},{1,2},{2,3}}
%e A369197                          {{3},{1,3},{2,3}}
%e A369197                          {{1,2},{1,3},{2,3}}
%o A369197 (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
%Y A369197 The minimal case is A000272.
%Y A369197 Connected case of A066383 and A369196, loopless A369192 and A369193.
%Y A369197 The loopless case is A129271, connected case of A369191.
%Y A369197 The case of equality is A368951, connected case of A368597.
%Y A369197 This is the connected case of A369194.
%Y A369197 A000085, A100861, A111924 count set partitions into singletons or pairs.
%Y A369197 A001187 counts connected graphs, unlabeled A001349.
%Y A369197 A006125 counts (simple) graphs, unlabeled A000088.
%Y A369197 A006129 counts covering graphs, unlabeled A002494.
%Y A369197 A054548 counts graphs covering n vertices with k edges, with loops A369199.
%Y A369197 A062740 counts connected loop-graphs.
%Y A369197 A322661 counts covering loop-graphs, unlabeled A322700.
%Y A369197 A368927 counts choosable loop-graphs, covering A369140.
%Y A369197 A369141 counts non-choosable loop-graphs, covering A369142.
%Y A369197 Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.
%K A369197 nonn
%O A369197 0,3
%A A369197 _Gus Wiseman_, Jan 17 2024
%E A369197 a(0) changed to 1 and a(7) onwards from _Andrew Howroyd_, Feb 02 2024