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

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 24 2003

Keywords

• nonn

Comments

a(n) <= A086875(n).

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.

Crossrefs

Cf. A086190, A086207, A086875.

Programs

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