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A083667
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Number of antisymmetric binary relations on a set of n labeled points.
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7
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1, 2, 12, 216, 11664, 1889568, 918330048, 1338925209984, 5856458868470016, 76848453272063549952, 3025216211508053707410432, 357271984146678126737757198336, 126579320351263180234426948827254784
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OFFSET
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0,2
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COMMENTS
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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
2*sum(1/A083667(n), n>=2) = 2*sum(2^(-n)*3^(-((n*(n-1))/2)), n>=2) = sum(1/prod(A008776(k), k=1..n), n>=1) = Sum(1/(Product(2*3^k, k=1...n)), n>=1) = 0.17609845431233461692099660022134. - Alexander R. Povolotsky, Aug 08 2011
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LINKS
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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.
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FORMULA
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a(n) = 3*a(n-1)^2/a(n-2). - Michael Somos, Sep 16 2005
a(n) = 2^n * 3^((n*(n-1))/2).
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MAPLE
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A083667:=n->2^n*3^((n^2-n)/2); seq(A083667(n), n=0..15); # Wesley Ivan Hurt, Nov 30 2013
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MATHEMATICA
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Table[2^n*3^((n^2-n)/2), {n, 0, 15}] (* Wesley Ivan Hurt, Nov 30 2013 *)
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PROG
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(GAP) a := n -> 2^n * 3^Binomial(n, 2);
(PARI) a(n)=2^n*3^((n^2-n)/2)
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CROSSREFS
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Cf. A083670.
Sequence in context: A182161 A165950 A208651 * A092124 A009525 A009683
Adjacent sequences: A083664 A083665 A083666 * A083668 A083669 A083670
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KEYWORD
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easy,nonn
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AUTHOR
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Goetz Pfeiffer (Goetz.Pfeiffer(AT)nuigalway.ie), May 02 2003
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EXTENSIONS
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Name simplified by Franklin T. Adams-Watters, Aug 07 2011
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STATUS
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approved
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