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%I #23 Apr 05 2020 20:21:48
%S 1,1,1,1,1,1,1,1,1,1,1,1,1,29937600,81729648000
%N Triangle read by rows: T(n,m) is the number of inequivalent n X m matrices under action of the Klein group, with one-tenth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n*m != 0 mod 10).
%C Computed using Polya s enumeration theorem for coloring.
%H María Merino, <a href="/A287384/b287384.txt">Rows n=0..32 of triangle, flattened</a>
%H 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 G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10)=(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=Sum_{i=1..10} x_i, y2=Sum_{i=1..10} x_i^2, and occurrences of numbers are ceiling(m*n/10) for the first k numbers and floor(m*n/10) for the last (10-k) numbers, if m*n = k mod 10.
%e For n = 4 and m = 3 the T(4,3)=29937600 solutions are colorings of 4 X 3 matrices in 10 colors inequivalent under the action of the Klein group with exactly 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 occurrences of each color (coefficient of x1^2 x2^2 x3^1 x4^1 x5^1 x6^1 x7^1 x8^1 x9^1 x10^1).
%e Triangle begins:
%e ==========================================
%e n\m | 0 1 2 3 4
%e ----|-------------------------------------
%e 0 | 1
%e 1 | 1 1
%e 2 | 1 1 1
%e 3 | 1 1 1 1
%e 4 | 1 1 1 29937600 81729648000
%Y Cf. A283435, A286892, A287020, A287021, A287022, A287377, A287378, A287383.
%K nonn,tabl
%O 0,14
%A _María Merino_, Imanol Unanue, May 24 2017
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