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A053764 a(n) = 3^(n^2 - n).

%I A053764 #42 Jul 02 2025 16:01:59
%S A053764 1,1,9,729,531441,3486784401,205891132094649,109418989131512359209,
%T A053764 523347633027360537213511521,22528399544939174411840147874772641,
%U A053764 8727963568087712425891397479476727340041449,30432527221704537086371993251530170531786747066637049,955004950796825236893190701774414011919935138974343129836853841,269721605590607563262106870407286853611938890184108047911269431464974473521
%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 Deléham_, Dec 09 2007
%C A053764 [1,9,729,531441,3486784401,...] is the Hankel transform of A005159. - _Philippe Deléham_, Dec 10 2007
%H A053764 Vincenzo Librandi, <a href="/A053764/b053764.txt">Table of n, a(n) for n = 0..46</a>
%H A053764 N. J. Fine and I. N. Herstein, <a href="http://projecteuclid.org/euclid.ijm/1255454112">The probability that a matrix be nilpotent</a>, Illinois J. Math., 2 (1958), 499-504.
%H A053764 Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H A053764 M. Gerstenhaber, <a href="http://projecteuclid.org/euclid.ijm/1255629831">On the number of nilpotent matrices with coefficients in a finite field</a>, 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 Deléham_, Aug 20 2005
%t A053764 Table[(3^(n^2 - n)), {n, 0, 20}] (* _Vincenzo Librandi_, Feb 24 2014 *)
%o A053764 (PARI) a(n) = 3^(n^2 - n); \\ _Joerg Arndt_, Feb 23 2014
%Y A053764 Cf. A053763, A380592.
%K A053764 easy,nonn
%O A053764 0,3
%A A053764 _Stephen G Penrice_, Mar 29 2000
%E A053764 More terms from _James Sellers_, Apr 08 2000