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%I #16 Jun 24 2018 16:00:46
%S 0,0,0,0,3,775,0,145,115100,68522769,1,4281,14051164,37460388596,
%T 97467398965031,3,115381,1608801153,20208371722051,257100007425866689,
%U 3363033541015148835823,20,2863227,180536313547,10980013072900632,691542997115450167856,45094635411084308447578413,3020745549854628001139950947779,136,68522707
%N Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly five 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=5. 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, A294792, A294793, A295197. T(n,1) is A056298.
%K nonn,tabl
%O 1,5
%A _Marko Riedel_, Nov 08 2017
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