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A294686 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry, 1 <= k <= n.

%I A294686 #35 Oct 05 2024 16:30:53
%S A294686 0,0,6,0,260,20720,6,5112,1223136,257706024,48,81876,67769552,
%T A294686 54278580036,44900438149488,260,1223396,3731753700,11681058472672,
%U A294686 38403264917970196,131160169581733489616,1200,17815020,207438938000,2570217454576416,33725471278376393424,460532748521625850986660,6467585568566200114362823920,5106,257706012,11681057249536,576229125971686224
%N A294686 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry, 1 <= k <= n.
%C A294686 Colors are not being permuted, i.e., Power Group Enumeration does not apply here.
%D A294686 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A294686 Andrew Howroyd, <a href="/A294686/b294686.txt">Table of n, a(n) for n = 1..820</a> (first 40 rows)
%H A294686 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>
%F A294686 T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=4 and S(n,k) Stirling numbers of the second kind.
%F A294686 T(n,k) = A184277(n,k) - 4*A184284(n,k) + 6*A184271(n,k) - 4. - _Andrew Howroyd_, Oct 05 2024
%e A294686 Triangle begins:
%e A294686     0;
%e A294686     0,       6;
%e A294686     0,     260,      20720;
%e A294686     6,    5112,    1223136,      257706024;
%e A294686    48,   81876,   67769552,    54278580036,    44900438149488;
%e A294686   260, 1223396, 3731753700, 11681058472672, 38403264917970196, 131160169581733489616;
%e A294686   ...
%o A294686 (PARI) T(n,m)=my(k=4); k!*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), k, 2) ))/(n*m) \\ _Andrew Howroyd_, Oct 05 2024
%Y A294686 Main diagonal is A376824.
%Y A294686 Cf. A294684, A294685, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056284.
%Y A294686 Cf. A184271, A184284, A184277.
%K A294686 nonn,tabl,nice
%O A294686 1,3
%A A294686 _Marko Riedel_, Nov 06 2017