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A294793 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry and swappable colors.

%I A294793 #16 Jun 24 2018 16:00:35
%S A294793 0,0,1,0,13,874,1,235,51075,10741819,2,3437,2823766,2261625725,
%T A294793 1870851589562,13,51275,155495153,486711524815,1600136051453135,
%U A294793 5465007068038102643,50,742651,8643289534,107092397450897,1405227969932349726,19188864521773558375127,269482732023591671431784330,221,10741763,486710971595,24009547064476683
%N A294793 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry and swappable colors.
%C A294793 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 A294793 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A294793 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>
%F A294793 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=4. 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 A294793 Cf. A294684, A294685, A294686, A294687, A294791, A294792, A294794, A295197. T(n,1) is A056297.
%K A294793 nonn,tabl
%O A294793 1,5
%A A294793 _Marko Riedel_, Nov 08 2017