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A294684 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry, 1 <= k <= n.

%I A294684 #56 Oct 05 2024 16:31:02
%S A294684 0,1,5,2,12,62,4,38,350,4154,6,106,2190,52486,1342206,12,360,14622,
%T A294684 699598,35792566,1908897150,18,1180,99878,9587578,981706830,
%U A294684 104715443850,11488774559742,34,4148,699250,134223974,27487816990,5864063066498,1286742755471398,288230376353050814
%N A294684 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry, 1 <= k <= n.
%C A294684 Colors are not being permuted, i.e., Power Group Enumeration does not apply here.
%D A294684 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A294684 Andrew Howroyd, <a href="/A294684/b294684.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%H A294684 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus</a>.
%H A294684 Marko Riedel, <a href="/A294684/a294684.maple.txt">Maple code for sequences A294684, A294685, A294686, A294687</a>.
%F A294684 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=2 and S(n,k) Stirling numbers of the second kind.
%F A294684 T(n,k) = A184271(n,k) - 2. - _Andrew Howroyd_, Oct 05 2024
%e A294684 Triangle begins:
%e A294684    0;
%e A294684    1,    5;
%e A294684    2,   12,    62;
%e A294684    4,   38,   350,    4154;
%e A294684    6,  106,  2190,   52486,   1342206;
%e A294684   12,  360, 14622,  699598,  35792566,   1908897150;
%e A294684   18, 1180, 99878, 9587578, 981706830, 104715443850, 11488774559742;
%e A294684   ...
%e A294684 For the 2 X 2 and two colors we find
%e A294684   +---+  +---+  +---+  +---+  +---+
%e A294684   |X| |  | |X|  |X| |  |X|X|  |X| |
%e A294684   +-+-+  +-+-+  +-+-+  +-+-+  +-+-+
%e A294684   | | |  |X|X|  | |X|  | | |  |X| |
%e A294684   +-+-+  +-+-+  +-+-+  +-+-+  +-+-+
%t A294684 With[{Q = 2}, Table[(Q!/n/k) Sum[Sum[EulerPhi[d] EulerPhi[f] StirlingS2[GCD[d, f] (n/d) (k/f), Q], {f, Divisors@ k}], {d, Divisors@ n}], {n, 8}, {k, n}]] // Flatten (* _Michael De Vlieger_, Nov 08 2017 *)
%o A294684 (PARI) T(n,m) = {2*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 2, 2) ))/(n*m)} \\ _Andrew Howroyd_, Oct 05 2024
%Y A294684 Main diagonal is A376822.
%Y A294684 Cf. A184271, A294685, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A052823.
%K A294684 nonn,tabl,nice
%O A294684 1,3
%A A294684 _Marko Riedel_, Nov 06 2017