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A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.

%I A294791 #33 Jun 24 2018 16:00:01
%S A294791 0,1,4,1,7,31,3,23,179,2107,3,55,1095,26271,671103,7,189,7327,350063,
%T A294791 17896831,954459519,9,595,49939,4794087,490853415,52357746895,
%U A294791 5744387279871,19,2101,349715,67115111,13743921631,2932032057731,643371380132743,144115188277194943,29,7315,2485591,954444607,390937468407,166799988703927,73201365371896619
%N A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.
%C A294791 Two colorings are equivalent if there is a permutation of the colors that takes one to the other in addition to translational symmetries on the torus. (Power Group Enumeration.)
%D A294791 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A294791 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>
%H A294791 Marko Riedel, <a href="/A294791/a294791_2.maple.txt">Maple code for sequences A294791, A294792, A294793, A294794.</a>
%F A294791 T(n,k) = (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma) with Q=2. The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.
%e A294791 For the 2 X 2 grid and two colors we find T(2,2) = 4:
%e A294791   +---+  +---+  +---+  +---+
%e A294791   |X| |  |X| |  |X|X|  |X| |
%e A294791   +-+-+  +-+-+  +-+-+  +-+-+
%e A294791   | | |  | |X|  | | |  |X| |
%e A294791   +-+-+  +-+-+  +-+-+  +-+-+
%Y A294791 Cf. A152175, A294684, A294685, A294686, A294687, A294792, A294793, A294794, A295197. T(n,1) is A056295.
%K A294791 nonn,tabl,nice
%O A294791 1,3
%A A294791 _Marko Riedel_, Nov 08 2017