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A002884
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Number of nonsingular n X n matrices over GF(2) (order of the group GL(n,2)); order of Chevalley group A_n (2); order of projective special linear group PSL_n(2).
(Formerly M4302 N1798)
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86
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1, 1, 6, 168, 20160, 9999360, 20158709760, 163849992929280, 5348063769211699200, 699612310033197642547200, 366440137299948128422802227200, 768105432118265670534631586896281600, 6441762292785762141878919881400879415296000
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OFFSET
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0,3
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COMMENTS
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Also number of bases for GF(2^n) over GF(2).
Also (apparently) number of n X n matrices over GF(2) having permanent = 1. - Hugo Pfoertner, Nov 14 2003
The previous comment is true because over GF(2) permanents and determinants are the same. - Joerg Arndt, Mar 07 2008
The number of automorphisms of (Z_2)^n (the direct product of n copies of Z_2). - Peter Eastwood, Apr 06 2015
Note that n! divides a(n) since the subgroup of GL(n,2) consisting of all permutation matrices is isomorphic to S_n (the n-th symmetric group). - Jianing Song, Oct 29 2022
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REFERENCES
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Carter, Roger W. Simple groups of Lie type. Pure and Applied Mathematics, Vol. 28. John Wiley & Sons, London-New York-Sydney, 1972. viii+331pp. MR0407163 (53 #10946). See page 2.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985, p. xvi.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 131.
Horadam, K. J., Hadamard matrices and their applications. Princeton University Press, Princeton, NJ, 2007. xiv+263 pp. See p. 132.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n) = Product_{i=0..n-1} (2^n-2^i).
a(n) = 2^(n*(n-1)/2) * Product_{i=1..n} (2^i - 1).
a(n) = (6*a(n-1)^2*a(n-3) - 8*a(n-1)*a(n-2)^2) / (a(n-2)*a(n-3)) for n > 2. - Seiichi Manyama, Oct 20 2016
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EXAMPLE
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PSL_2(2) is isomorphic to the symmetric group S_3 of order 6.
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MAPLE
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# First program
A002884:= n-> product(2^n - 2^i, i=0..n-1);
# Second program
A002884:= n-> 2^(n*(n-1)/2) * product( 2^i - 1, i=1..n);
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MATHEMATICA
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Table[Product[2^n-2^i, {i, 0, n-1}], {n, 0, 13}] (* Harvey P. Dale, Aug 07 2011 *)
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PROG
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(Magma) [1] cat [(&*[2^n -2^k: k in [0..n-1]]): n in [1..15]]; // G. C. Greubel, Aug 31 2023
(SageMath) [product(2^n -2^j for j in range(n)) for n in range(16)] # G. C. Greubel, Aug 31 2023
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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