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%I #28 Apr 29 2019 04:45:53
%S 1,1,1,1,1,1,1,1,90,5712,1,1,1260,416064,168168000,1,60,28440,
%T 42045600,76385194200,155840192585280,1,180,415800,3216282300,
%U 31168037156256,342718542439257600,3574641463338838464000,1,630,8408400,320818773240,14181456923282880,794364769671213312000,40694019408428534970822000,2416738787895064029335795945088
%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-fifth of 1s, 2s, 3s, 4s and 5s (ordered occurrences rounded up/down if n*m != 0 mod 5).
%C Computed using Polya's enumeration theorem for coloring.
%H María Merino, <a href="/A287021/b287021.txt">Rows n=0..38 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)=(y1^(m*n) + 3*y2^(m*n/2))/4 for even n and m;
%F (y1^(m*n) + y1^n*y2^((m*n-m)/2) + 2*y2^(m*n/2))/4 for odd n and even m;
%F (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..5} x_i, y2=Sum_{i=1..5} x_i^2, and occurrences of numbers are ceiling(m*n/5) for the first k numbers and floor(m*n/5) for the last (5-k) numbers, if m*n = k mod 5.
%e For n = 5 and m = 2 the T(5,2) = 28440 solutions are colorings of 5 X 2 matrices in 5 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 x5^2).
%e Triangle begins:
%e ============================================================
%e n\m | 0 1 2 3 4 5
%e ----|-------------------------------------------------------
%e 0 | 1
%e 1 | 1 1
%e 2 | 1 1 1
%e 3 | 1 1 90 5712
%e 4 | 1 1 1260 416064 168168000
%e 5 | 1 60 28440 42045600 76385194200 155840192585280
%Y Cf. A283435, A286892, A287020, A287022, A287377, A287378, A287383, A287384.
%K nonn,tabl
%O 0,9
%A _María Merino_, Imanol Unanue, May 18 2017
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