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A053763 a(n) = 2^(n^2 - n). 30
1, 1, 4, 64, 4096, 1048576, 1073741824, 4398046511104, 72057594037927936, 4722366482869645213696, 1237940039285380274899124224, 1298074214633706907132624082305024, 5444517870735015415413993718908291383296, 91343852333181432387730302044767688728495783936 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Nilpotent n X n matrices over GF(2). Also number of simple digraphs (without self-loops) on n labeled nodes (see also A002416)

For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(4) (sequence A053291). - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001

(-1)^ceil(n/2) * resultant of the Chebyshev polynomial of first kind of degree n and Chebyshev polynomial of first kind of degree (n+1) (cf. A039991). - Benoit Cloitre, Jan 26 2003

The number of reflexive binary relations on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005

Contribution from Rick L. Shepherd, Dec 24 2008: (Start)

Number of gift exchange scenarios where, for each person k of n people,

i) k gives gifts to g(k) of the others, where 0 <= g(k) <= n-1,

ii) k gives no more than one gift to any specific person,

iii) k gives no single gift to two or more people and

iv) there is no other person j such that j and k jointly give a single gift.

(In other words -- but less precisely -- each person k either gives no gifts or gives exactly one gift per person to 1 <= g(k) <= n-1 others.) (End)

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.

J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 521.

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 5, Eq. (1.1.5).

A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..35

G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184 (see Th. 3).

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.

FORMULA

Sequence given by the Hankel transform (see A001906 for definition) of A059231 = {1, 1, 5, 29, 185, 1257, 8925, 65445, 491825, ...}; example : det([1, 1, 5, 29; 1, 5, 29, 185; 5, 29, 185, 1257; 29, 185, 1257, 8925]) = 4^6 = 4096. - Philippe Deléham, Aug 20 2005

a(n)=4^(C(2+n,n)), n>=-2. - Zerinvary Lajos, Jun 16 2007

a(n) = Sum_{i=0..n^2-n} C(n^2-n, i). [Rick L. Shepherd, Dec 24 2008]

EXAMPLE

a(2)=4 because there are four 2 x 2 nilpotent matrices over GF(2):{{0,0},{0,0}},{{0,1},{0,0}},{{0,0},{1,0}},{{1,1,},{1,1}}  where 1+1=0. - Geoffrey Critzer, Oct 05 2012

MAPLE

seq(4^(binomial(2+n, n)), n=-2..11); - Zerinvary Lajos, Jun 16 2007

a:=n->mul (4^j, j=1..n): seq(a(n), n=-1..12); - Zerinvary Lajos, Oct 03 2007

MATHEMATICA

Table[2^(2*Binomial[n, 2]), {n, 0, 20}] (* Geoffrey Critzer, Oct 04 2012 *)

PROG

(PARI) a(n)=1<<(n^2-n) \\ Charles R Greathouse IV, Nov 20 2012

CROSSREFS

Cf. A053773, A006125, A000273, A000984, A002416.

Sequence in context: A088065 A053718 A053773 * A193611 A193755 A194501

Adjacent sequences:  A053760 A053761 A053762 * A053764 A053765 A053766

KEYWORD

easy,nonn,nice

AUTHOR

Stephen G. Penrice (spenrice(AT)ets.org), Mar 29 2000

STATUS

approved

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Last modified October 23 16:00 EDT 2014. Contains 248468 sequences.