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A287020
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Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-fourth each of 1s, 2s, 3s and 4s (ordered occurrences rounded up/down if n*m != 0 mod 4).
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9
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1, 1, 1, 1, 1, 6, 1, 1, 46, 1926, 1, 12, 648, 92544, 15767640, 1, 30, 6312, 3943710, 2933201376, 2061379857600, 1, 90, 92400, 192994200, 577186150464, 1605824110657800, 5363188066566330000, 1, 318, 1051140, 10266445476, 118129589107200, 1340797019145183600
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OFFSET
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0,6
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COMMENTS
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Computed using Polya's enumeration theorem for coloring.
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LINKS
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María Merino, Rows n=0..42 of triangle, flattened
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).
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FORMULA
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G.f.: g(x1,x2,x3,x4)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
(y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
(y1^(m*n) + y1^m*y2^((m*n-n)/2) + 2*y2^(m*n/2))/4 for even n and odd m;
(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+x4, y2=x1^2+x2^2+x3^2+x4^2, and occurrences of numbers are ceiling(m*n/4) for the first k numbers and floor(m*n/4) for the last (4-k) numbers, if m*n = k mod 4.
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EXAMPLE
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For n = 4 and m = 2 the T(4,2) = 648 solutions are colorings of 4 X 2 matrices in 4 colors inequivalent under the action of the Klein group with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2).
Triangle begins:
========================================================
n\m | 0 1 2 3 4 5
----|---------------------------------------------------
0 | 1
1 | 1 1
2 | 1 1 6
3 | 1 1 46 1926
4 | 1 12 648 92544 15767640
5 | 1 30 6312 3943710 2933201376 2061379857600
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CROSSREFS
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Cf. A283435, A286892, A287021, A287022, A287377, A287378, A287383, A287384.
Sequence in context: A156764 A156765 A015117 * A172375 A075377 A046792
Adjacent sequences: A287017 A287018 A287019 * A287021 A287022 A287023
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KEYWORD
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nonn,tabl
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AUTHOR
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María Merino, Imanol Unanue, May 18 2017
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STATUS
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approved
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