A348958 Triangular array read by rows. T(n,k) = A002884(n)/A002884(n-k)*2^((n-k)(n-k-1)), n>=0, 0<=k<=n.

1, 1, 1, 4, 6, 6, 64, 112, 168, 168, 4096, 7680, 13440, 20160, 20160, 1048576, 2031616, 3809280, 6666240, 9999360, 9999360, 1073741824, 2113929216, 4095737856, 7679508480, 13439139840, 20158709760, 20158709760, 4398046511104, 8727373545472, 17182016667648, 33290157293568, 62419044925440, 109233328619520, 163849992929280, 163849992929280

Offset: 0

Author

Geoffrey Critzer, Nov 04 2021

Keywords

• nonn
• tabl

Comments

Let ~ be the equivalence relation on the set of n X n matrices over GF(2) defined by A ~ B if and only if the dimension of the image of A^n is equal to the dimension of the image of B^n. Let A be a recurrent matrix (Cf A348622) of rank k. Then T(n,k) is the size of the equivalence class containing A.

Let X_n be the random variable that assigns to each n X n matrix A over GF(q) the value j = nullity(A^n). Then limit as n->oo of P(X_n = j) = Product_{i>=1}(1 - 1/q^i)*q^(j^2-j)/|GL_j(F_q)|. - Geoffrey Critzer, Jan 02 2025

Examples

			Triangle begins:
  1,
  1,       1,
  4,       6,       6,
  64,      112,     168,     168,
  4096,    7680,    13440,   20160,   20160,
  1048576, 2031616, 3809280, 6666240, 9999360, 9999360
		

Crossrefs

Cf. A348622, A002884 (main diagonal), A053763 (column k=0).

Programs

Mathematica

R[n_, d_] := Product[q^n - q^i, {i, 0, n - 1}]/Product[q^(n - d) - q^i, {i, 0, n - d - 1}];Table[Table[R[n, d] q^((n - d) (n - d - 1)), {d, 0, n}], {n, 0,10}] // Grid

Formula

T(n,k) = A002884(n)/A002884(n-k)*2^((n-k)*(n-k-1)).

Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/B(n) = f(x)*g(y*x) where f(x) = Sum_{n>=0} q^(n^2-n)*x^n/B(n), g(x) = Sum_{n>=0} Product_{i=0..n-1} (q^n-q^i)x^n/B(n), B(n) = Product_{i=0..n-1} (q^n-q^i)/(q-1)^n and q=2. - Geoffrey Critzer, Jan 02 2025