%I #55 Mar 19 2020 16:22:37

%S 1,1,1,1,2,1,1,6,3,1,1,12,16,4,1,1,34,84,31,5,1,1,87,579,318,52,6,1,1,

%T 302,5721,5366,867,81,7,1,1,1118,87381,155291,28722,2028,118,8,1,1,

%U 5478,2104349,7855628,1919895,115391,4251,165,9,1,1,32302,78315231,675054876,250530482,14662562,393963,8214,222,10,1

%N Triangle read by rows: T(n,k) is the number of simple graphs on n unlabeled nodes having chromatic number k, 1 <= k <= n.

%C T(n,1) = T(n,n) = 1 (here we count the empty graph and the complete graph). T(n,n-1) = n-1 (here we count the graphs with clique number equal to n-1). - _Geoffrey Critzer_, Oct 12 2016

%C Row sums give A000088. - _Joerg Arndt_, Oct 13 2016

%H Keith Briggs, <a href="http://keithbriggs.info/cgt.html">combinatorial graph theory</a>, see entry "number of graphs on n nodes with clique number k".

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000098">The chromatic number of a graph.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticNumber.html">Chromatic Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/n-ChromaticGraph.html">n-Chromatic Graph</a>

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 2, 1;

%e 1, 6, 3, 1;

%e 1, 12, 16, 4, 1;

%e 1, 34, 84, 31, 5, 1;

%e 1, 87, 579, 318, 52, 6, 1;

%e 1, 302, 5721, 5366, 867, 81, 7, 1;

%e 1, 1118, 87381, 155291, 28722, 2028, 118, 8, 1;

%e 1, 5478, 2104349, 7855628, 1919895, 115391, 4251, 165, 9, 1;

%e ...

%o (Sage) # prints triangle with a leading zero in each row

%o for n in range(1, 8) :

%o st = [0 for j in range(n+1)]

%o G = graphs(n)

%o for g in G :

%o st[ g.chromatic_number() ] += 1

%o print(st)

%o # _Joerg Arndt_, Oct 13 2016

%Y Columns k=1..9 are A057427, A076278, A076279, A076280, A076281, A076282, A076283, A205567, A205568.

%Y Partial row sums include A033995, A076315, A076316, A076317, A076318, A076319, A076320, A076321.

%Y Row sums are A000088.

%Y Cf. A084269 (connected), A115597 (essentially the same sequence).

%K nonn,tabl

%O 1,5

%A _Eric W. Weisstein_, May 24 2003

%E Offset corrected by _Joerg Arndt_, Oct 13 2016

%E a(36)-a(55) from _Joerg Arndt_, Oct 15 2016

%E a(56)-a(66) from _Andrew Howroyd_, Dec 02 2018