A286526 Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4, with a fifth of 1's, 2's, 3's, 4's and 5's (ordered occurrences rounded up/down if n^2 != 0 mod 5).

1, 1, 1, 2874, 84086160, 77920099694640, 1787320731699689472000, 1208369393947533515948886636000, 22022604563875220592723146462014970246400, 10631042739086498005729294276105510004209560426195000, 136864426940639977623403211038729959780835360788855628470904385280

Offset

0,4

Links

María Merino, Table of n, a(n) for n = 0..37

M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Formula

G.f.: g(x1,x2,x3,x4,x5) = 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..5} x_i, y2 = Sum_{i=1..5} x_i^2, y4 = Sum_{i=1..5} x_i^4 and occurrences of numbers are ceiling(n^2/5) for the first k numbers and floor(n^2/5) for the last (5-k) numbers, if n^2 = k mod 5.

Example

For n=3 the a(3)=2874 solutions are colorings of 3 X 3 matrices in 5 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 2, 2, 1 of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^1).

Crossrefs

Cf. A054752, A082963, A286447, A286525.

Sequence in context: A254580 A254355 A261859 * A081427 A101999 A066174

Adjacent sequences: A286523 A286524 A286525 * A286527 A286528 A286529

Keyword

nonn

Author

María Merino, Imanol Unanue, May 11 2017

Status

approved