cp's OEIS Frontend

A372172 Number of labeled simple graphs on n vertices with exactly one triangle.

%I A372172 #9 Aug 01 2024 01:21:34
%S A372172 0,0,0,1,16,290,6980,235270,11298056,777154308,76560083040
%N A372172 Number of labeled simple graphs on n vertices with exactly one triangle.
%C A372172 The unlabeled version is A372194.
%F A372172 Binomial transform of A372171.
%e A372172 The a(4) = 16 graphs:
%e A372172   12,13,23
%e A372172   12,14,24
%e A372172   13,14,34
%e A372172   23,24,34
%e A372172   12,13,14,23
%e A372172   12,13,14,24
%e A372172   12,13,14,34
%e A372172   12,13,23,24
%e A372172   12,13,23,34
%e A372172   12,14,23,24
%e A372172   12,14,24,34
%e A372172   12,23,24,34
%e A372172   13,14,23,34
%e A372172   13,14,24,34
%e A372172   13,23,24,34
%e A372172   14,23,24,34
%t A372172 cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
%t A372172 Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cys[#]]==1&]],{n,0,5}]
%Y A372172 For no triangles we have A213434, covering A372168 (unlabeled A372169).
%Y A372172 Column k = 1 of A372170, unlabeled A263340.
%Y A372172 The covering case is A372171, unlabeled A372174.
%Y A372172 For all cycles (not just triangles) we have A372193, covering A372195.
%Y A372172 The unlabeled version is A372194.
%Y A372172 A001858 counts acyclic graphs, unlabeled A005195.
%Y A372172 A006125 counts simple graphs, unlabeled A000088.
%Y A372172 A006129 counts covering graphs, unlabeled A002494
%Y A372172 A054548 counts labeled covering graphs by edges, unlabeled A370167.
%Y A372172 A372167 counts covering graphs by triangles, unlabeled A372173.
%Y A372172 Cf. A000272, A105784, A121251, A137916, A236570, A372176.
%K A372172 nonn,more
%O A372172 0,5
%A A372172 _Gus Wiseman_, Apr 24 2024
%E A372172 a(8)-a(10) from _Andrew Howroyd_, Aug 01 2024