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 A053763 a(n) = 2^(n^2 - n). 34
 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 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 T. D. Noe, Table of n, a(n) for n = 0..35 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. M. Gerstenhaber, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., Vol. 5 (1961), 330-333. A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82. G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, 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. 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^binomial(2+n, 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 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,changed AUTHOR Stephen G. Penrice (spenrice(AT)ets.org), Mar 29 2000 STATUS approved

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Last modified May 23 12:38 EDT 2019. Contains 323514 sequences. (Running on oeis4.)