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%I #13 Oct 18 2023 09:22:09
%S 1,2,14,352,32512,11239424,14761852928,74524125036544,
%T 1459094811012235264,111539381955990155952128,
%U 33460660604316425324211470336,39542320578630779599776165929156608,184615341335916919478531491782548361576448
%N Number of n X n matrices over GF(2) whose characteristic polynomial is a product of (not necessarily distinct) linear factors, i.e., the characteristic polynomial has the form x^k(1+x)^(n-k) for some 0 <= k <= n.
%H L. Kaylor and D. Offner, <a href="https://doi.org/10.2140/involve.2014.7.627">Counting matrices over a finite field with all eigenvalues in the field</a>, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645.
%F Sum_{n>=0} a(n)*x^n/A002884(n) = (Sum_{n>=0} A053763(n)x^n/A002884(n))^2 = (Product_{n>=1} 1/(1-x/2^n))^2.
%e a(2) = 14 because there are 16 2 X 2 matrices over GF(2) and all except {{0,1},{1,1}} and {{1,1},{1,0}} have characteristic polynomials of the desired form.
%t nn = 12; q = 2; Table[Product[q^n - q^i, {i, 0, n - 1}], {n, 0, nn}] CoefficientList[Series[Product[1/(1 - u/q^r), {r, 1, \[Infinity]}]^2, {u, 0, nn}], u]
%Y Cf. A002884, A053763.
%K nonn
%O 0,2
%A _Geoffrey Critzer_, Jul 10 2021
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