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A106512 Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).

%I A106512 #22 Sep 11 2019 12:30:25
%S A106512 0,0,0,0,2,0,0,6,0,0,0,12,6,2,0,0,20,24,18,0,0,0,30,60,84,30,2,0,0,42,
%T A106512 120,260,240,66,0,0,0,56,210,630,1020,732,126,2,0,0,72,336,1302,3120,
%U A106512 4100,2184,258,0,0,0,90,504,2408,7770,15630,16380,6564,510,2,0,0,110
%N A106512 Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).
%C A106512 Note that we keep one edge in the circular graph even when there's only one node (so there are 0 colorings of one node with k colors).
%C A106512 Number of closed walks of length n on the complete graph K_{k}. - _Andrew Howroyd_, Mar 12 2017
%H A106512 Andrew Howroyd, <a href="/A106512/b106512.txt">Table of n, a(n) for n = 1..1274</a>
%F A106512 a(n, k) = (k-1)^n + (-1)^n * (k-1).
%e A106512 From _Andrew Howroyd_, Mar 12 2017: (Start)
%e A106512 Table begins:
%e A106512   0 0   0     0      0       0        0        0         0 ...
%e A106512   0 2   6    12     20      30       42       56        72 ...
%e A106512   0 0   6    24     60     120      210      336       504 ...
%e A106512   0 2  18    84    260     630     1302     2408      4104 ...
%e A106512   0 0  30   240   1020    3120     7770    16800     32760 ...
%e A106512   0 2  66   732   4100   15630    46662   117656    262152 ...
%e A106512   0 0 126  2184  16380   78120   279930   823536   2097144 ...
%e A106512   0 2 258  6564  65540  390630  1679622  5764808  16777224 ...
%e A106512   0 0 510 19680 262140 1953120 10077690 40353600 134217720 ...
%e A106512 (End)
%e A106512 a(4,3) = 18 because there are three choices for the first node's color (call it 1) and then two choices for the second node's color (call it 2) and then the remaining two nodes can be 12, 13, or 32. So in total there are 3*2*3 = 18 ways. a(3,4) = 4*3*2 = 24 because the three nodes must be three distinct colors.
%Y A106512 Columns include A092297, A226493. Main diagonal is A118537.
%Y A106512 Rows 2-7 are A002378, A007531, A091940, A061167, A131472, A133499.
%Y A106512 Cf. A090860, A208535.
%K A106512 nonn,tabl
%O A106512 1,5
%A A106512 _Joshua Zucker_, May 29 2005
%E A106512 a(67) corrected by _Andrew Howroyd_, Mar 12 2017