A297892 Triangle read by rows. T(n,k) is the number of n X n diagonalizable matrices over GF(3) that have rank k, 0 <= k <= n, n >= 0.

1, 1, 2, 1, 24, 14, 1, 234, 1638, 236, 1, 2160, 147420, 254880, 12692, 1, 19602, 12349260, 208173240, 124394292, 1783784, 1, 176904, 1011404394, 157378969440, 916910326332, 157779262368, 811523288

Offset: 0

Author

Geoffrey Critzer, Jan 07 2018

Keywords

• nonn
• tabl

Examples

			Triangle begins
  1;
  1,     2;
  1,    24,       14;
  1,   234,     1638,       236;
  1,  2160,   147420,    254880,     12692;
  1, 19602, 12349260, 208173240, 124394292, 1783784;
		

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.

Crossrefs

Cf. A296548, A053846 (main diagonal), A290516 (row sums).

Programs

Mathematica

nn = 5; g[n_] := (q - 1)^n  q^Binomial[n, 2] FunctionExpand[QFactorial[n, q]] /. q -> 3;G[n] := Sum[u z^r/g[r], {r, 0, nn}]; Grid[Map[Select[#, # > 0 &] &,Table[g[n], {n, 0, nn}] CoefficientList[Series[Sum[(u z)^r/g[r] , {r, 0, nn}]^2 Sum[
       z^r/g[r], {r, 0, nn}], {z, 0, nn}], {z, u}]]]

Formula

T(n,k)/A053290(n) is the coefficient of y^k*x^n in the expansion of Sum_{n>=0} x^n\A053290(n) * (Sum_{n>=0} y*x^n\A053290(n))^2.