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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.

%I A348958 #20 Jan 03 2025 10:56:20
%S A348958 1,1,1,4,6,6,64,112,168,168,4096,7680,13440,20160,20160,1048576,
%T A348958 2031616,3809280,6666240,9999360,9999360,1073741824,2113929216,
%U A348958 4095737856,7679508480,13439139840,20158709760,20158709760,4398046511104,8727373545472,17182016667648,33290157293568,62419044925440,109233328619520,163849992929280,163849992929280
%N 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.
%C A348958 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.
%C A348958 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
%F A348958 T(n,k) = A002884(n)/A002884(n-k)*2^((n-k)*(n-k-1)).
%F A348958 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
%e A348958 Triangle begins:
%e A348958   1,
%e A348958   1,       1,
%e A348958   4,       6,       6,
%e A348958   64,      112,     168,     168,
%e A348958   4096,    7680,    13440,   20160,   20160,
%e A348958   1048576, 2031616, 3809280, 6666240, 9999360, 9999360
%t A348958 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
%Y A348958 Cf. A348622, A002884 (main diagonal), A053763 (column k=0).
%K A348958 nonn,tabl
%O A348958 0,4
%A A348958 _Geoffrey Critzer_, Nov 04 2021