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A053764
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a(n) = 3^(n^2 - n).
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12
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1, 1, 9, 729, 531441, 3486784401, 205891132094649, 109418989131512359209, 523347633027360537213511521, 22528399544939174411840147874772641, 8727963568087712425891397479476727340041449, 30432527221704537086371993251530170531786747066637049, 955004950796825236893190701774414011919935138974343129836853841, 269721605590607563262106870407286853611938890184108047911269431464974473521
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OFFSET
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0,3
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COMMENTS
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Number of nilpotent n X n matrices X over GF(3), that is, the number of n X n matrices X over GF(3) satisfying X^k = 0 for some k >= 1.
More generally, Fine and Herstein prove that the probability that an n X n matrix over GF(p^m) is nilpotent is 1/p^(mn) and the probability that an n X n matrix over Z/mZ is nilpotent is 1/k^n, where k is the product of the distinct prime factors of m.
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LINKS
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FORMULA
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Sequence given by the Hankel transform (see A001906 for definition) of A082181 = {1, 1, 10, 109, 1270, 15562, 198100, ...}; example: det([1, 1, 10, 109; 1, 10, 109, 1270; 10, 109, 1270, 15562; 109, 1270, 15562, 198100]) = 9^6 = 531441. - Philippe Deléham, Aug 20 2005
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MATHEMATICA
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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