%I A053764
%S A053764 1,1,9,729,531441,3486784401,205891132094649,109418989131512359209,523347633027360537213511521,
%T A053764 22528399544939174411840147874772641,8727963568087712425891397479476727340041449,
%U A053764 30432527221704537086371993251530170531786747066637049,9550049507968252368931907017744140119199351389743431298\
               36853841,269721605590607563262106870407286853611938890184108047911269431464974473521
%N A053764 a(n) = 3^(n^2 - n)
%C A053764 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.
%C A053764 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.
%C A053764 Is this the same sequence (apart from the initial term) as A053854? - 
               Philippe DELEHAM, Dec 09 2007
%C A053764 [1,9,729,531441,3486784401,...] is the Hankel transform of A005159. - 
               Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 10 2007
%D A053764 N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, 
               Illinois J. Math., 2 (1958), 499-504.
%D A053764 M. Gerstenhaber, On the number of nilpotent matrices with coefficients 
               in a finite field. Illinois J. Math., Vol. 5 (1961), 330-333.
%F A053764 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
%p A053764 with(finance):seq(mul(futurevalue( 1, 2, n),k=0..n),n=-1..10); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2008
%Y A053764 Cf. A053763.
%Y A053764 Sequence in context: A069034 A053847 A053854 this_sequence A122251 A015481 
               A145183
%Y A053764 Adjacent sequences: A053761 A053762 A053763 this_sequence A053765 A053766 
               A053767
%K A053764 easy,nonn
%O A053764 0,3
%A A053764 Stephen G. Penrice (spenrice(AT)ets.org), Mar 29 2000
%E A053764 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Apr 08 2000