A346210 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.

1, 2, 14, 352, 32512, 11239424, 14761852928, 74524125036544, 1459094811012235264, 111539381955990155952128, 33460660604316425324211470336, 39542320578630779599776165929156608, 184615341335916919478531491782548361576448

Offset

0,2

Links

Table of n, a(n) for n=0..12.

L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645.

Formula

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.

Example

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.

Mathematica

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]

Crossrefs

Cf. A002884, A053763.

Sequence in context: A094155 A132626 A101524 * A255493 A013039 A011809

Adjacent sequences: A346207 A346208 A346209 * A346211 A346212 A346213

Keyword

nonn

Author

Geoffrey Critzer, Jul 10 2021

Status

approved