Search: id:A053763
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%I A053763
%S A053763 1,1,4,64,4096,1048576,1073741824,4398046511104,72057594037927936,
%T A053763 4722366482869645213696,1237940039285380274899124224,1298074214633706907132624082305024,
%U A053763 5444517870735015415413993718908291383296,91343852333181432387730302044767688728495783936
%N A053763 a(n) = 2^(n^2 - n).
%C A053763 Nilpotent n X n matrices over GF(2). Also number of labeled digraphs 
               on n nodes.
%C A053763 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
%C A053763 (-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 (benoit7848c(AT)orange.fr), Jan 26 
               2003
%C A053763 The number of reflexive binary relations on an n-element set. - Justin 
               Witt (justinmwitt(AT)gmail.com), Jul 12 2005
%C A053763 Contribution from Rick L. Shepherd (rshepherd2(AT)hotmail.com), Dec 24 
               2008: (Start)
%C A053763 Number of gift exchange scenarios where, for each person k of n people,
%C A053763 i) k gives gifts to g(k) of the others, where 0 <= g(k) <= n-1,
%C A053763 ii) k gives no more than one gift to any specific person,
%C A053763 iii) k gives no single gift to two or more people and
%C A053763 iv) there is no other person j such that j and k jointly give a single 
               gift.
%C A053763 (In other words -- but less precisely -- each person k either gives no 
               gifts or
%C A053763 gives exactly one gift per person to 1 <= g(k) <= n-1 others.) (End)
%D A053763 N. J. Fine and I. N. Herstein, The probability that a matrix be nilpotent, 
               Illinois J. Math., 2 (1958), 499-504.
%D A053763 M. Gerstenhaber, On the number of nilpotent matrices with coefficients 
               in a finite field. Illinois J. Math., Vol. 5 (1961), 330-333.
%D A053763 F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 
               1973, p. 5, Eq. (1.1.5).
%D A053763 J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 
               2004; p. 521.
%D A053763 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%H A053763 T. D. Noe, <a href="b053763.txt">Table of n, a(n) for n=0..35</a>
%H A053763 G. Kuperberg, <a href="http://arXiv.org/abs/math.CO/0008184">Symmetry 
               classes of alternating-sign matrices under one roof, arXiv math.CO/
               0008184</a> (see Th. 3).
%H A053763 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/">Counting 
               Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), 
               Article 04.3.2.
%F A053763 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 DELEHAM (kolotoko(AT)wanadoo.fr), 
               Aug 20 2005
%F A053763 a(n)=4^(C(2+n,n)), n>=-2 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 16 2007
%F A053763 a(n) = Sum_{i=0..n^2-n} C(n^2-n, i) [From Rick L. Shepherd (rshepherd2(AT)hotmail.com), 
               Dec 24 2008]
%p A053763 seq(4^(binomial(2+n,n)), n=-2..11); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Jun 16 2007
%p A053763 a:=n->mul (4^j,j=1..n): seq(a(n),n=-1..12); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Oct 03 2007
%p A053763 with(finance):seq(mul(futurevalue( 1, 1, n),k=0..n),n=- 1..12); - Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jul 01 2008
%Y A053763 Cf. A053773, A006125, A000273.
%Y A053763 Cf. A000984.
%Y A053763 Sequence in context: A088065 A053718 A053773 this_sequence A053923 A051191 
               A120581
%Y A053763 Adjacent sequences: A053760 A053761 A053762 this_sequence A053764 A053765 
               A053766
%K A053763 easy,nonn,nice
%O A053763 0,3
%A A053763 Stephen G. Penrice (spenrice(AT)ets.org), Mar 29 2000

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