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%I #26 Apr 05 2020 21:37:14
%S 1,1,1,1,40864828320,7792009289281728000,
%T 187746872107299580970294400000,
%U 614005731326101652800803825889630961295360,176445174659483893854948844253232539237396497554309120000,7090469783239448892319287907564531885316857076509137838529329991091840000
%N Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4, with one-tenth 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^2 != 0 mod 10).
%C Computed using Polya's enumeration theorem for coloring.
%H María Merino, <a href="/A287261/b287261.txt">Table of n, a(n) for n = 0..29</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) = (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..10} x_i, y2=Sum_{i=1..10} x_i^2, y4=Sum_{i=1..10} x_i^4 and occurrences of numbers are ceiling(n^2/10) for the first k numbers and floor(n^2/10) for the last (10-k) numbers, if n^2 = k mod 10.
%e For n = 3 the a(4) = 40864828320 solutions are colorings of 4 X 4 matrices in 10 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2 x7^1 x8^1 x9^1 x10^1).
%Y Cf. A286397, A082963, A286447, A286525, A286526, A287239, A287245, A287249, A287250.
%K nonn
%O 0,5
%A _María Merino_, Imanol Unanue, May 22 2017