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A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.

%I A286331 #28 Aug 29 2019 15:21:59
%S A286331 1,1,1,1,9,6,1,49,294,168,1,225,7350,37800,20160,1,961,144150,4036200,
%T A286331 19373760,9999360,1,3969,2542806,326932200,8543828160,39687459840,
%U A286331 20158709760,1,16129,42677334,23435953128,2812314375360,71124337751040,325139829719040,163849992929280
%N A286331 Triangle read by rows: T(n,k) is the number of n X n matrices of rank k over F_2.
%H A286331 Robert Israel, <a href="/A286331/b286331.txt">Table of n, a(n) for n = 0..1710</a> (rows 0 to 57, flattened)
%H A286331 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A286331 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A286331 Wikipedia, <a href="http://en.wikipedia.org/wiki/Q-binomial">q-binomial</a>
%F A286331 T(n,k) = Product_{j=0..k-1} (2^n - 2^j)^2/(2^k - 2^j) = A022166(n,k) * Product_{j=0..k-1} (2^n - 2^j).
%e A286331 Triangle T(n,k) begins:
%e A286331   1;
%e A286331   1,   1;
%e A286331   1,   9,      6;
%e A286331   1,  49,    294,     168;
%e A286331   1, 225,   7350,   37800,    20160;
%e A286331   1, 961, 144150, 4036200, 19373760, 9999360;
%e A286331   ...
%e A286331 T(2,1) = 9 because there are 9, 2 X 2 matrices in F_2 that have rank 1: {{0, 0}, {0, 1}}, {{0, 0}, {1, 0}}, {{0, 0}, {1, 1}}, {{0, 1}, {0, 0}}, {{0, 1}, {0, 1}}, {{1, 0},  {0, 0}}, {{1, 0}, {1, 0}}, {{1,1}, {0, 0}}, {{1, 1}, {1, 1}}.
%p A286331 T:= (n,k) -> mul((2^n-2^j)^2/(2^k-2^j),j=0..k-1):
%p A286331 seq(seq(T(n,k),k=0..n),n=0..10); # _Robert Israel_, May 15 2017
%t A286331 q = 2; Table[Table[Product[(q^n - q^i)^2/(q^k - q^i), {i, 0, k - 1}], {k, 0, n}], {n, 0, 6}] // Grid
%Y A286331 Main diagonal is A002884.
%Y A286331 Column for k = 1 is A060867.
%Y A286331 Row sums are A002416.
%K A286331 nonn,tabl
%O A286331 0,5
%A A286331 _Geoffrey Critzer_, May 07 2017