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A053763 a(n) = 2^(n^2 - n). 66
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)^ceiling(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
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)
In general, sequences of the form m^((n^2 - n)/2) enumerate the graphs with n labeled nodes with m types of edge. a(n) therefore is the number of labeled graphs with n nodes with 4 types of edge. To clarify the comment from Benoit Cloitre, dated Jan 26 2003, in this context: simple digraphs (without self-loops) have four types of edge. These types of edges are as follows: the absent edge, the directed edge from A -> B, the directed edge from B -> A and the bidirectional edge, A <-> B. - Mark Stander, Apr 11 2019
REFERENCES
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).
LINKS
Marcus Brinkmann, Extended Affine and CCZ Equivalence up to Dimension 4, Ruhr University Bochum (2019).
N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, Illinois J. Math., 2 (1958), 499-504.
Murray Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., Vol. 5 (1961), 330-333.
Antal Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Greg Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Annals of mathematics, Second Series, Vol. 156, No. 3 (2002), pp. 835-866, arXiv preprint, arXiv:math/0008184 [math.CO], 2000-2001 (see Th. 3).
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
Götz 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^binomial(n, n-2). - Zerinvary Lajos, Jun 16 2007
a(n) = Sum_{i=0..n^2-n} binomial(n^2-n, i). - Rick L. Shepherd, Dec 24 2008
G.f. A(x) satisfies: A(x) = 1 + x * A(4*x). - Ilya Gutkovskiy, Jun 04 2020
Sum_{n>=1} 1/a(n) = A319016. - Amiram Eldar, Oct 27 2020
Sum_{n>=0} a(n)*u^n/A002884(n) = Product_{r>=1} 1/(1-u/q^r). - Geoffrey Critzer, Oct 28 2021
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(n, n-2)), n=0..12); # Zerinvary Lajos, Jun 16 2007
a:=n->mul(4^j, j=1..n-1): seq(a(n), n=0..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
Row sums of A123554, A189898, A346412, A346214.
Sequence in context: A088065 A053718 A053773 * A193611 A193755 A194501
KEYWORD
easy,nonn,nice
AUTHOR
Stephen G Penrice, Mar 29 2000
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)