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A086699
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Number of n X n matrices over GF(2) with rank n-1.
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2
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1, 9, 294, 37800, 19373760, 39687459840, 325139829719040, 10654345790226432000, 1396491759480328106803200, 732164571206732295657278668800, 1535460761275478347250381697633484800
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OFFSET
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1,2
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COMMENTS
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a(n)/2^(n^2) is the probability that a random linear operator T on an n dimensional vector space over the field with two elements is such that the dimension of the range of T equals n-1. This probability is Product{j>=2} 1 - 1/2^j which is 2 times the probability that the dimension of the range of T equals n. Cf. A048651. - Geoffrey Critzer, Jun 28 2017
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LINKS
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FORMULA
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for n>=2 : a(n) = product j=0...n-2 (2^n - 2^j)^2 / (2^(n-1)- 2^j).
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MATHEMATICA
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Table[Product[(q^n - q^i)^2/(q^(n - 1) - q^i), {i, 0, n - 2}] /. q -> 2, {n, 0, 15}] (* Geoffrey Critzer, Jun 28 2017 *)
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PROG
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(PARI) a(n) = prod(j=0, n-2, (2^n - 2^j)^2 / (2^(n-1)- 2^j)); \\ Michel Marcus, Jun 28 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 28 2003
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EXTENSIONS
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STATUS
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approved
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