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

1, 2, 12, 216, 11664, 1889568, 918330048, 1338925209984, 5856458868470016, 76848453272063549952, 3025216211508053707410432, 357271984146678126737757198336, 126579320351263180234426948827254784

Offset: 0

Author

Goetz Pfeiffer (Goetz.Pfeiffer(AT)nuigalway.ie), May 02 2003

Keywords

• easy
• nonn

Comments

Hankel transform of aeration of A110520. - Paul Barry, Sep 15 2009

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

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

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..50
• Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
• Jeffrey R. Goodwin, The 3x+1 Problem and Integer Representations, arXiv:1504.03040 [math.NT], 2015.
• G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
• G. Wirsching, On the combinatorial structure of 3N+1 predecessor sets, Discrete Mathematics, Vol. 148 (1996), 265-286.

Crossrefs

Cf. A083670.

Programs

GAP

a := n -> 2^n * 3^Binomial(n, 2);

Maple

A083667:=n->2^n*3^((n^2-n)/2); seq(A083667(n), n=0..15); # Wesley Ivan Hurt, Nov 30 2013

Mathematica

Table[2^n*3^((n^2-n)/2), {n, 0, 15}] (* Wesley Ivan Hurt, Nov 30 2013 *)

PARI

a(n)=2^n*3^((n^2-n)/2)

Formula

a(n) = 3*a(n-1)^2/a(n-2). - Michael Somos, Sep 16 2005

a(n) = 2^n * 3^((n*(n-1))/2).

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

Extensions

Name simplified by Franklin T. Adams-Watters, Aug 07 2011