cp's OEIS Frontend

A083667 Number of antisymmetric binary relations on a set of n labeled points.

%I A083667 #71 Aug 24 2024 01:53:59
%S A083667 1,2,12,216,11664,1889568,918330048,1338925209984,5856458868470016,
%T A083667 76848453272063549952,3025216211508053707410432,
%U A083667 357271984146678126737757198336,126579320351263180234426948827254784
%N A083667 Number of antisymmetric binary relations on a set of n labeled points.
%C A083667 Hankel transform of aeration of A110520. - _Paul Barry_, Sep 15 2009
%C A083667 The number of infinite sets per level n in the Collatz Tree partitioning the inverse iterates of the number m, where the value n is given by the number of odds in sequences which converge to m in the 3x+1 Problem, and m is a positive odd integer, not divisible by 3, for which the 3x+1 Problem holds. We get the entire Collatz (3x+1) Tree for the case m = 1. Otherwise, we get a portion of the tree. We can think of the infinite sets as residue classes modulo powers of 3. In this way Wirsching gets an abstract description of the Collatz Tree along with its underlying combinatorial structure. See Corollary 3.2 in Wirsching's paper. - _Jeffrey R. Goodwin_, Jul 26 2011
%C A083667 Let T_n denote the n X n matrix with T_n(i,j) = 3^(min(i,j)-1); then a(n) = det(T_(n+1)). - _Lechoslaw Ratajczak_, May 11 2021
%C A083667 The number of simple directed graphs (digraphs) on n labeled vertices, allowing for loops but restricting each pair of distinct vertices to have at most one directed edge between them, in either direction. These constraints define a structure where each vertex can have a loop, and for any two distinct vertices, there are three possible relationships: no edge, a directed edge from the first vertex to the second, or a directed edge from the second vertex to the first. - _Constantinos Kourouzides_, Mar 25 2024
%H A083667 Vincenzo Librandi, <a href="/A083667/b083667.txt">Table of n, a(n) for n = 0..50</a>
%H A083667 Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
%H A083667 Jeffrey R. Goodwin, <a href="http://arxiv.org/abs/1504.03040">The 3x+1 Problem and Integer Representations</a>, arXiv:1504.03040 [math.NT], 2015.
%H A083667 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A083667 G. Wirsching, <a href="http://dx.doi.org/10.1016/0012-365X(94)00243-C">On the combinatorial structure of 3N+1 predecessor sets</a>, Discrete Mathematics, Vol. 148 (1996), 265-286.
%F A083667 a(n) = 3*a(n-1)^2/a(n-2). - _Michael Somos_, Sep 16 2005
%F A083667 a(n) = 2^n * 3^((n*(n-1))/2).
%F A083667 2*Sum_{n>=2} 1/a(n) = 2*Sum_{n>=2} 2^(-n)*3^(-((n*(n-1))/2)) = Sum_{n>=1} 1/Product_{k=1..n} A008776(k) = Sum_{n>=1} 1/Product_{k=1..n} 2*3^k = 0.1760984543123346169209966002213.... - _Alexander R. Povolotsky_, Aug 08 2011
%p A083667 A083667:=n->2^n*3^((n^2-n)/2); seq(A083667(n), n=0..15); # _Wesley Ivan Hurt_, Nov 30 2013
%t A083667 Table[2^n*3^((n^2-n)/2), {n, 0, 15}] (* _Wesley Ivan Hurt_, Nov 30 2013 *)
%o A083667 (GAP) a := n -> 2^n * 3^Binomial(n, 2);
%o A083667 (PARI) a(n)=2^n*3^((n^2-n)/2)
%Y A083667 Cf. A083670.
%K A083667 easy,nonn
%O A083667 0,2
%A A083667 Goetz Pfeiffer (Goetz.Pfeiffer(AT)nuigalway.ie), May 02 2003
%E A083667 Name simplified by _Franklin T. Adams-Watters_, Aug 07 2011