A343097 Array read by antidiagonals: T(n,k) is the number of k-colorings of an n X n grid, up to rotations and reflections.

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, 6, 120, 34960, 5398083, 4211744, 1, 0, 1, 7, 231, 252375, 537157696, 105918450471, 8590557312, 1, 0, 1, 8, 406, 1284066, 19076074375, 140738033618944, 18761832172500795, 70368882591744, 1, 0

Offset: 0

Author

Andrew Howroyd, Apr 14 2021

Keywords

• nonn
• tabl

Examples

			Array begins:
====================================================================
n\k | 0 1       2            3               4                 5
----+---------------------------------------------------------------
  0 | 1 1       1            1               1                 1 ...
  1 | 0 1       2            3               4                 5 ...
  2 | 0 1       6           21              55               120 ...
  3 | 0 1     102         2862           34960            252375 ...
  4 | 0 1    8548      5398083       537157696       19076074375 ...
  5 | 0 1 4211744 105918450471 140738033618944 37252918396015625 ...
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..860

Crossrefs

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

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

Main diagonal is A001326.

Cf. A246106, A343095, A343875.

Programs

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}

Formula

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.