cp's OEIS Frontend

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

%I A053722 #51 Apr 25 2024 09:11:33
%S A053722 1,4,22,316,6976,373024,32252032,6619979776,2253838544896,
%T A053722 1810098020122624,2442718932612677632,7758088894129169760256,
%U A053722 41674675294431186817908736,526370120583359572695165435904,11281778621698661853306239290703872
%N A053722 Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)).
%C A053722 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.
%C A053722 Conjecture: the two matrices I and 0 have the largest number of square roots. Checked for n=1..5. - _Alexey Slizkov_, Jan 11 2024
%D A053722 Vladeta Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.
%H A053722 Robert Israel, <a href="/A053722/b053722.txt">Table of n, a(n) for n = 1..81</a>
%H A053722 Jason Fulman and C. Ryan Vinroot, <a href="http://arxiv.org/abs/1306.0031">Generating functions for real character degree sums of finite general linear and unitary groups</a>, arXiv:1306.0031 [math.GR], 2013 (see Theorem 3.4 for a g.f.).
%H A053722 Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%F A053722 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]
%p A053722 Q:= Product(1+u/2^i,i=1..infinity)/Product(1-u^2/2^i,i=1..infinity):
%p A053722 S:= series(Q,u,31):
%p A053722 seq(coeff(S,u,n)*mul(2^i-1,i=1..n), n=1..30); # _Robert Israel_, Mar 26 2018
%t A053722 QP = QPochhammer; Q = (1-x) QP[-x, 1/2]/QP[x^2, 1/2];
%t A053722 Table[(-1)^n QP[2, 2, n] SeriesCoefficient[Q, {x, 0, n}], {n, 1, 14}] (* _Jean-François Alcover_, Sep 17 2018, from Maple *)
%o A053722 (SageMath)
%o A053722 g = lambda n: GL(n,2).order() if n>0 else 1
%o A053722 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
%o A053722 map(a053722, range(25))
%o A053722 # _Dmitrii Pasechnik_, Oct 02 2015
%Y A053722 Cf. A053725, A063505, A225371.
%K A053722 nonn
%O A053722 1,2
%A A053722 _Vladeta Jovovic_, Mar 23 2000