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A053722 Number of n X n binary matrices of order dividing 2 (also number of solutions to X^2=I in GL(n,2)). 25
1, 4, 22, 316, 6976, 373024, 32252032, 6619979776, 2253838544896, 1810098020122624, 2442718932612677632, 7758088894129169760256, 41674675294431186817908736, 526370120583359572695165435904 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

REFERENCES

V. Jovovic, The cycle index polynomials of some classical groups, Belgrade, 1995, unpublished.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

FORMULA

a(n) = sum k=1...[n/2] (2^n - 1)(2^n - 2) ... (2^n - 2^{n-k+1})/(2^k - 1)(2^k - 2)....(2^k - 2^{k-1}) * (2^{n-k} - 1)(2^{n-k} - 2)...(2^{n-k} - 2^{n-2k+1}). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 05 2001

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-Fran├žois Alcover, Sep 17 2018, from Maple *)

PROG

# (Sage)

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

CROSSREFS

Sequence in context: A119009 A326883 A317803 * A336212 A276122 A145504

Adjacent sequences:  A053719 A053720 A053721 * A053723 A053724 A053725

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, Mar 23 2000

STATUS

approved

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Last modified September 19 21:34 EDT 2020. Contains 337184 sequences. (Running on oeis4.)