cp's OEIS Frontend

A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.

%I A294687 #33 Oct 05 2024 16:30:49
%S A294687 0,0,0,0,300,92680,0,15750,13794150,8221452750,24,510312,1686135376,
%T A294687 4495236798162,11696087875731720,300,13794450,193054017440,
%U A294687 2425003938178050,30852000867277668428,403564024914127655401650,2400,343501500,21664357535320,1317601563731383350,82985159653854019928352,5411356249329837891442095560
%N A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.
%C A294687 Colors are not being permuted, i.e., Power Group Enumeration does not apply here.
%D A294687 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A294687 Andrew Howroyd, <a href="/A294687/b294687.txt">Table of n, a(n) for n = 1..325</a> (first 25 rows).
%H A294687 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus.</a>
%F A294687 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=5 and S(n,k) Stirling numbers of the second kind.
%e A294687 Triangle begins:
%e A294687    0;
%e A294687    0,      0;
%e A294687    0,    300,      92680;
%e A294687    0,  15750,   13794150,    8221452750;
%e A294687   24, 510312, 1686135376, 4495236798162, 11696087875731720;
%e A294687   ...
%o A294687 (PARI) T(n,m)=my(k=5); 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 A294687 Main diagonal is A376825.
%Y A294687 Cf. A294684, A294685, A294686, A294791, A294792, A294793, A294794. T(n,1) is A056285.
%K A294687 nonn,tabl,nice
%O A294687 1,5
%A A294687 _Marko Riedel_, Nov 06 2017