%I #17 Jun 24 2018 16:00:21

%S 0,0,3,1,18,345,2,136,7254,447156,5,946,158355,29032254,5647919665,18,

%T 7324,3580802,1961010826,1143822046786,694881637942816,43,56450,

%U 82968843,136166703562,238244961999013,434202285631866206,813943290958393433377,126,447138,1960981598,9651082393912,50656925726930746,276966813318877426118,1557582240509759704455566

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

%C 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 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

%H Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>

%F 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=3. 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.

%Y Cf. A294684, A294685, A294686, A294687, A294791, A294793, A294794, A295197. T(n,1) is A056296.

%K nonn,tabl

%O 1,3

%A _Marko Riedel_, Nov 08 2017