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Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.
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%I #11 May 09 2021 19:26:27

%S 1,1,0,1,1,0,1,2,1,0,1,3,6,1,0,1,4,21,102,1,0,1,5,55,2862,8548,1,0,1,

%T 6,120,34960,5398083,4211744,1,0,1,7,231,252375,537157696,

%U 105918450471,8590557312,1,0,1,8,406,1284066,19076074375,140738033618944,18761832172500795,70368882591744,1,0

%N Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

%H Andrew Howroyd, <a href="/A343097/b343097.txt">Table of n, a(n) for n = 0..860</a>

%F T(n,k) = (k^(n^2) + 2*k^((n^2 + 3*(n mod 2))/4) + k^((n^2 + (n mod 2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n + n mod 2)/2) )/8.

%e Array begins:

%e ====================================================================

%e n\k | 0 1 2 3 4 5

%e ----+---------------------------------------------------------------

%e 0 | 1 1 1 1 1 1 ...

%e 1 | 0 1 2 3 4 5 ...

%e 2 | 0 1 6 21 55 120 ...

%e 3 | 0 1 102 2862 34960 252375 ...

%e 4 | 0 1 8548 5398083 537157696 19076074375 ...

%e 5 | 0 1 4211744 105918450471 140738033618944 37252918396015625 ...

%e ...

%o (PARI) T(n,k) = {(k^(n^2) + 2*k^((n^2 + 3*(n%2))/4) + k^((n^2 + (n%2))/2) + 2*k^(n*(n+1)/2) + 2*k^(n*(n+n%2)/2) )/8}

%Y Rows 0..5 are A000012, A001477, A002817, A217331, A217338, A283033.

%Y Columns 0..10 are A000007, A000012, A054247, A054739, A054751, A054752, A286392, A286393, A286394, A286396, A286397.

%Y Main diagonal is A001326.

%Y Cf. A246106, A343095, A343875.

%K nonn,tabl

%O 0,8

%A _Andrew Howroyd_, Apr 14 2021