A053722 Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)).

1, 4, 22, 316, 6976, 373024, 32252032, 6619979776, 2253838544896, 1810098020122624, 2442718932612677632, 7758088894129169760256, 41674675294431186817908736, 526370120583359572695165435904, 11281778621698661853306239290703872

Offset: 1

Author

Vladeta Jovovic, Mar 23 2000

Keywords

• nonn

Comments

In characteristic 2, A^2 = I if and only if B^2 = 0 where B = I + A, so a(n) is also equal to the number of n X n binary matrices B such that B^2 = 0.

Conjecture: the two matrices I and 0 have the largest number of square roots. Checked for n=1..5. - Alexey Slizkov, Jan 11 2024

References

• Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.

Links

• Robert Israel, Table of n, a(n) for n = 1..81
• Jason Fulman and C. Ryan Vinroot, Generating functions for real character degree sums of finite general linear and unitary groups, arXiv:1306.0031 [math.GR], 2013 (see Theorem 3.4 for a g.f.).
• Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

Crossrefs

Cf. A053725, A063505, A225371.

Programs

Maple

Q:= Product(1+u/2^i,i=1..infinity)/Product(1-u^2/2^i,i=1..infinity):
S:= series(Q,u,31):
seq(coeff(S,u,n)*mul(2^i-1,i=1..n), n=1..30); # Robert Israel, Mar 26 2018

Mathematica

QP = QPochhammer; Q = (1-x) QP[-x, 1/2]/QP[x^2, 1/2];
Table[(-1)^n QP[2, 2, n] SeriesCoefficient[Q, {x, 0, n}], {n, 1, 14}] (* Jean-François Alcover, Sep 17 2018, from Maple *)

SageMath

g = lambda n: GL(n,2).order() if n>0 else 1
a053722 = lambda n: g(n)*sum(1/(g(k)*g(n-2*k)*2**(k**2+2*k*(n-2*k))) for k in range(1+floor(n/2))) if n>0 else 0
map(a053722, range(25))
# Dmitrii Pasechnik, Oct 02 2015

Formula

a(n) = Sum_{k=0..floor(n/2)} (2^n - 1)(2^{n-1} - 1) ... (2^{n-2k+1}-1) * 2^{k(k-1)/2} / ((2^k - 1)(2^{k-1} - 1) ... (2^1 - 1)). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001 [Corrected using the paper by Morrison, which also mentions that there is an error in this entry. k = 0 contributes 1 to the sum. If omitted, this gives the number of matrices of order exactly 2. Jan Kristian Haugland, Apr 24 2024]