login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified December 22 22:37 EST 2014. Contains 252372 sequences.