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A294794
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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.
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8
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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
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OFFSET
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1,5
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COMMENTS
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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.)
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REFERENCES
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F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
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LINKS
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Table of n, a(n) for n=1..30.
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
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FORMULA
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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.
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CROSSREFS
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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
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KEYWORD
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nonn,tabl
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AUTHOR
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Marko Riedel, Nov 08 2017
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STATUS
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approved
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