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A286892 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-third each of 1s, 2s and 3s (ordered occurrences rounded up/down if m*n != 0 mod 3).

%I A286892 #34 Apr 29 2019 04:45:39
%S A286892 1,1,1,1,1,3,1,3,27,438,1,6,140,8766,504504,1,16,1056,189774,33258880,
%T A286892 6573403050,1,48,8730,4292514,2366403930,1387750992012,
%U A286892 846182953495152,1,108,63108,99797220,159511561440,282061024690536,530143167401850960,976645996512669379710
%N A286892 Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-third each of 1s, 2s and 3s (ordered occurrences rounded up/down if m*n != 0 mod 3).
%C A286892 Computed using Polya's enumeration theorem for coloring.
%H A286892 María Merino, <a href="/A286892/b286892.txt">Rows n=0..47 of triangle, flattened</a>
%H A286892 M. Merino and I. Unanue, <a href="https://doi.org/10.1387/ekaia.17851">Counting squared grid patterns with Pólya Theory</a>, EKAIA, 34 (2018), 289-316 (in Basque).
%F A286892 G.f.: g(x1,x2,x3)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
%F A286892 (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
%F A286892 (y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
%F A286892 (y1^(m*n) + y1^n*y2^((m*n-n)/2) + y1^m*y2^((m*n-m)/2) + y1*y2^((m*n-1)/2))/4 for odd n and m; where coefficient correspond to y1=x1+x2+x3, y2=x1^2+x2^2+x3^2, and occurrences of numbers are ceiling(m*n/3) for  the first k numbers and floor(m*n/3) for the last (3-k) numbers, if m*n = k mod 3.
%e A286892 For n = 3 and m = 2 the T(3,2) = 27 solutions are colorings of 3 X 2 matrices in 3 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2).
%e A286892 Triangle begins:
%e A286892 =================================================
%e A286892 n\m | 0    1   2      3       4         5
%e A286892 ----|--------------------------------------------
%e A286892 0   | 1
%e A286892 1   | 1    1
%e A286892 2   | 1    1   3
%e A286892 3   | 1    3   27     438
%e A286892 4   | 1    6   140    8766    504504
%e A286892 5   | 1    16  1056   189774  33258880   6573403050
%Y A286892 Cf. A283435, A287020, A287021, A287022, A287377, A287378, A287383, A287384.
%K A286892 nonn,tabl
%O A286892 0,6
%A A286892 _María Merino_, Imanol Unanue, May 15 2017