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 A294794 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. 8
 0, 0, 0, 0, 3, 775, 0, 145, 115100, 68522769, 1, 4281, 14051164, 37460388596, 97467398965031, 3, 115381, 1608801153, 20208371722051, 257100007425866689, 3363033541015148835823, 20, 2863227, 180536313547, 10980013072900632, 691542997115450167856, 45094635411084308447578413, 3020745549854628001139950947779, 136, 68522707 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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.) REFERENCES F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973. LINKS Marko Riedel et al., Burnside lemma and translational symmetries of the torus. FORMULA 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. CROSSREFS Cf. A294684, A294685, A294686, A294687, A294791, A294792, A294793, A295197. T(n,1) is A056298. Sequence in context: A172895 A259369 A259371 * A293252 A341567 A287695 Adjacent sequences:  A294791 A294792 A294793 * A294795 A294796 A294797 KEYWORD nonn,tabl AUTHOR Marko Riedel, Nov 08 2017 STATUS approved

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Last modified August 13 20:49 EDT 2022. Contains 356107 sequences. (Running on oeis4.)