A086098 Sum of rank(M) over all n X n matrices over GF(2).
1, 21, 1141, 208965, 139889701, 354550756581, 3464730268306021, 131934922593867875685, 19707939574875773323508581, 11599530748705611712884878698341, 26983642577843418550426409405086580581, 248652621703069011230281370429818425958461285
Offset: 1
Keywords
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..50
- Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Programs
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PARI
a(n) = {my(q=2); sum(r=1, n, r*prod(j=0, r-1, (q^n-q^j)^2/(q^r-q^j)))} \\ Andrew Howroyd, Jul 08 2018
Formula
For prime power q the number of rank-r n X n matrices over GF(q) is F(r, n) = product j=0..(r-1) (q^n-q^j)^2/(q^r-q^j) so a(n) = sum r=1..n r*product j=0..(r-1) (q^n-q^j)^2/(q^r-q^j) . In this case q=2.
a(n) = Sum_{r=1..n} r*Product_{j=0, r-1} (2^n - 2^j)^2/(2^r - 2^j). - Andrew Howroyd, Jul 08 2018
Extensions
Terms a(8) and beyond from Andrew Howroyd, Jul 08 2018
Comments