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

1, 1, 9, 729, 531441, 3486784401, 205891132094649, 109418989131512359209, 523347633027360537213511521, 22528399544939174411840147874772641, 8727963568087712425891397479476727340041449, 30432527221704537086371993251530170531786747066637049, 955004950796825236893190701774414011919935138974343129836853841, 269721605590607563262106870407286853611938890184108047911269431464974473521

Offset: 0

Author

Stephen G Penrice, Mar 29 2000

Keywords

• easy
• nonn

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 Deléham, Dec 09 2007

[1,9,729,531441,3486784401,...] is the Hankel transform of A005159. - Philippe Deléham, Dec 10 2007

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..46
• N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
• Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.
• M. Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., Vol. 5 (1961), 330-333.

Crossrefs

Cf. A053763, A380592.

Programs

Mathematica

Table[(3^(n^2 - n)), {n, 0, 20}] (* Vincenzo Librandi, Feb 24 2014 *)

PARI

a(n) = 3^(n^2 - n); \\ Joerg Arndt, Feb 23 2014

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 Deléham, Aug 20 2005

Extensions

More terms from James Sellers, Apr 08 2000