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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). 9
1, 1, 1, 1, 1, 3, 1, 3, 27, 438, 1, 6, 140, 8766, 504504, 1, 16, 1056, 189774, 33258880, 6573403050, 1, 48, 8730, 4292514, 2366403930, 1387750992012, 846182953495152, 1, 108, 63108, 99797220, 159511561440, 282061024690536, 530143167401850960, 976645996512669379710 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Computed using Polya's enumeration theorem for coloring.

LINKS

María Merino, Rows n=0..47 of triangle, flattened

FORMULA

G.f.: g(x1,x2,x3)=(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, 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.

EXAMPLE

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).

Triangle begins:

=================================================

n\m | 0    1   2      3       4         5

----|--------------------------------------------

0   | 1

1   | 1    1

2   | 1    1   3

3   | 1    3   27     438

4   | 1    6   140    8766    504504

5   | 1    16  1056   189774  33258880   6573403050

CROSSREFS

Cf. A283435, A287020, A287021, A287022, A287377, A287378, A287383, A287384.

Sequence in context: A040173 A320952 A128777 * A215828 A067009 A229755

Adjacent sequences:  A286889 A286890 A286891 * A286893 A286894 A286895

KEYWORD

nonn,tabl

AUTHOR

María Merino, Imanol Unanue, May 15 2017

STATUS

approved

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Last modified April 19 04:19 EDT 2019. Contains 322237 sequences. (Running on oeis4.)