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A372173 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).

%I A372173 #16 Dec 29 2024 19:18:50
%S A372173 1,0,1,1,1,4,1,1,0,1,7,5,4,2,2,1,0,1,0,0,1,24,16,23,12,15,8,7,4,4,1,3,
%T A372173 1,1,1,0,0,1,0,0,0,1,69,79,113,103,105,83,73,58,45,34,31,22,14,16,10,
%U A372173 4,8,5,2,3,2,2,2,1,0,1,1,0,0,0,1,0,0,0,0,1
%N A372173 Irregular triangle read by rows where T(n,k) is the number of unlabeled simple graphs covering n vertices with exactly k triangles, 0 <= k <= binomial(n,3).
%H A372173 Andrew Howroyd, <a href="/A372173/b372173.txt">Table of n, a(n) for n = 0..340</a> (rows 0..10)
%H A372173 Gus Wiseman, <a href="/A372173/a372173.png">All unlabeled simple graphs covering 5 vertices, grouped by number of triangles</a>.
%e A372173 Triangle begins:
%e A372173   1
%e A372173   0
%e A372173   1
%e A372173   1 1
%e A372173   4 1 1 0 1
%e A372173   7 5 4 2 2 1 0 1 0 0 1
%Y A372173 Row sums are A002494, labeled A006129.
%Y A372173 Row lengths are A050407.
%Y A372173 The non-covering version is A263340, labeled A372170.
%Y A372173 Counting edges instead of triangles gives A370167, labeled A054548.
%Y A372173 The labeled version is A372167.
%Y A372173 Column k = 0 is A372169, labeled A372168 (non-covering A213434).
%Y A372173 Column k = 1 is A372174, labeled A372171.
%Y A372173 Column k = 1 is also the covering case of A372194, labeled A372172.
%Y A372173 A000088 counts unlabeled graphs, labeled A006125.
%Y A372173 A001858 counts acyclic graphs, unlabeled A005195.
%Y A372173 A372176 counts labeled graphs by directed cycles, covering A372175.
%Y A372173 Cf. A000055, A053530, A137917, A137918, A140637, A144958, A322700, A370169.
%K A372173 nonn,tabf
%O A372173 0,6
%A A372173 _Gus Wiseman_, Apr 23 2024
%E A372173 a(21) onwards from _Andrew Howroyd_, Dec 29 2024