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A294685
Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.
9
0, 0, 9, 2, 91, 2022, 9, 738, 43315, 2679246, 30, 5613, 950062, 174184755, 33887517990, 91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814, 258, 338259, 497812638, 816999710223, 1429469771994078, 2605213713043722909, 4883659745750360600262, 729, 2679228, 11765822365, 57906482267826, 303941554100145501
OFFSET
1,3
COMMENTS
Colors are not being permuted, i.e., Power Group Enumeration does not apply here.
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
LINKS
FORMULA
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=3 and S(n,k) Stirling numbers of the second kind.
T(n,k) = A184284(n,k) - 3*A184271(n,k) + 3. - Andrew Howroyd, Oct 05 2024
EXAMPLE
Triangle begins:
0;
0, 9;
2, 91, 2022;
9, 738, 43315, 2679246;
30, 5613, 950062, 174184755, 33887517990;
91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814;
...
PROG
(PARI) T(n, m)=6*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d, e), 3, 2) ))/(n*m) \\ Andrew Howroyd, Oct 05 2024
CROSSREFS
Main diagonal is A376823.
Sequence in context: A174837 A096042 A038292 * A200238 A368982 A254666
KEYWORD
nonn,tabl,nice
AUTHOR
Marko Riedel, Nov 06 2017
STATUS
approved