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%I #23 Apr 29 2019 05:41:49
%S 1,1,1,5688,504508320,2029169127793680,333772217080092664473600,
%T 1966297518276227170017585421188600,
%U 474436367892839446541884570454351985506872320,4529567636413022031420100639004131328550592354551163392000,1664947024157601976065851576560401128416782438266187161307818265349050000
%N Number of inequivalent n X n matrices over an alphabet of size 6 under action of dihedral group of the square D_4, with one-sixth each of 1s, 2s, 3s, 4s, 5s and 6s (ordered occurrences rounded up/down if n^2 != 0 mod 6).
%C Computed using Polya's enumeration theorem for coloring.
%H María Merino, <a href="/A287239/b287239.txt">Table of n, a(n) for n = 0..34</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)=1/8*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..6} x_i, y2=Sum_{i=1..6} x_i^2, y4=Sum_{i=1..6} x_i^4 and occurrences of numbers are ceiling(n^2/6) for the first k numbers and floor(n^2/6) for the last (6-k) numbers, if n^2 = k mod 6.
%e For n = 3 the a(3) = 5688 solutions are colorings of 3 X 3 matrices in 6 colors inequivalent under the action of D_4 with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2).
%Y Cf. A286392, A082963, A286447, A286525, A286526.
%K nonn
%O 0,4
%A _María Merino_, Imanol Unanue, May 22 2017
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