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A296605
Rectangle read by rows: T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have exactly k distinct eigenvalues, n >= 0, 0 <= k <= 3.
2
1, 0, 0, 0, 0, 3, 0, 0, 0, 3, 36, 0, 0, 3, 702, 1404, 0, 3, 38070, 379080, 0, 3, 5351346, 341368830, 0, 3, 2434569858, 1231457092866, 0, 3, 2987199920970, 17481694843567584, 0, 3, 11966842794993066, 1077553466091961763220
OFFSET
0,6
LINKS
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
FORMULA
T(n,k)/A053290(n) is the coefficient of y^(3-k)*x^n in the expansion of (-1 + y + Sum_{n>=0} x^n/A053290(n))^3.
EXAMPLE
Array begins:
1, 0, 0, 0,
0, 3, 0, 0,
0, 3, 36, 0,
0, 3, 702, 1404,
0, 3, 38070, 379080,
0, 3, 5351346, 341368830
MATHEMATICA
nn = 8; g[ n_] := (q - 1)^n q^Binomial[n, 2] FunctionExpand[
QFactorial[n, q]] /. q -> 3; G[u_, z_] := Sum[z^k/\[Gamma][k], {k, 0, nn}] - 1 + u ; Grid[Map[Reverse, Table[\[Gamma][n], {n, 0, nn}] CoefficientList[Series[G[u, z]^3, {z, 0, nn}], {z, u}]]]
CROSSREFS
Cf. A290516 (row sums).
Sequence in context: A336642 A151671 A267502 * A354488 A278478 A364107
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Dec 16 2017
STATUS
approved