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A053764
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a(n) = 3^(n^2 - n)
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10
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1, 1, 9, 729, 531441, 3486784401, 205891132094649, 109418989131512359209, 523347633027360537213511521, 22528399544939174411840147874772641, 8727963568087712425891397479476727340041449, 30432527221704537086371993251530170531786747066637049, 955004950796825236893190701774414011919935138974343129836853841, 269721605590607563262106870407286853611938890184108047911269431464974473521
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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.
Is this the same sequence (apart from the initial term) as A053854? - Philippe DELEHAM, Dec 09 2007
[1,9,729,531441,3486784401,...] is the Hankel transform of A005159. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2007
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REFERENCES
| N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
M. Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field. Illinois J. Math., Vol. 5 (1961), 330-333.
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FORMULA
| 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 DELEHAM (kolotoko(AT)wanadoo.fr), Aug 20 2005
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MAPLE
| with(finance):seq(mul(futurevalue( 1, 2, n), k=0..n), n=-1..10); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2008
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CROSSREFS
| Cf. A053763.
Sequence in context: A069034 A053847 A053854 * A122251 A015481 A185274
Adjacent sequences: A053761 A053762 A053763 * A053765 A053766 A053767
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KEYWORD
| easy,nonn
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AUTHOR
| Stephen G. Penrice (spenrice(AT)ets.org), Mar 29 2000
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 08 2000
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