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A006125
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2^(n(n-1)/2).
(Formerly M1897)
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88
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1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736, 35184372088832, 36028797018963968, 73786976294838206464, 302231454903657293676544, 2475880078570760549798248448, 40564819207303340847894502572032
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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Number of graphs on n labeled nodes; also number of outcomes of labeled n-team round-robin tournaments.
Number of perfect matchings of order n Aztec diamond [see Speyer]
Number of Gelfand-Zeitlin patterns with bottom row [1,2,3,...,n]. [Zeilberger]
For n >= 1 a(n) is the size of the Sylow 2-subgroup of the Chevalley group A_n(2) (sequence A002884) - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 30 2001
Comment from Jim Propp (propp(AT)math.wisc.edu): a(n) is the number of ways to tile the region
.........o-----o
.........|.....|
......o--o.....o--o
......|...........|
...o--o...........o--o
...|.................|
o--o.................o--o
|.......................|
|.......................|
|.......................|
o--o.................o--o
...|.................|
...o--o...........o--o
......|...........|
......o--o.....o--o
.........|.....|
.........o-----o
(top-to-bottom distance = 2n) with dominoes like either of
o--o o-----o
|..| |.....|
|..| o-----o
|..|
o--o
The number of domino tilings in A006253, A004003, A006125 is the number of perfect matchings in the relevant graphs. There are results of Jockusch and Ciucu that if a planar graph has a rotational symmetry then the number of perfect matchings is a square or twice a square - this applies to these 3 sequences. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 12 2001
Let M_n denotes the n X n matrix with M_n(i,j)=binomial(2i,j); then det(M_n)=a(n+1). - Benoit Cloitre, Apr 21 2002
Smallest power of 2 which can be expressed as the product of n distinct numbers (powers of 2), e.g. a(4) = 1024 = 2*4*8*16. Also smallest number which can be expressed as the product of n distinct powers. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Nov 10 2002
The number of binary relations that are both reflexive and symmetric on an n-element set. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. Then the probability of winning for the first time after n tails is A005329 / A006125 . The probability of having won before n+1 tails is A114604 / A006125 . - Joshua Zucker, Dec 14 2005
a(n)=A126883(n-1)+1. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 12 2007
Equals right border of triangle A158474(unsigned). [From Gary W. Adamson, Mar 20 2009]
a(n-1) is the number of simple labeled graphs on n nodes such that every node has even degree. -Geoffrey Critzer, Oct 21 2011.
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REFERENCES
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Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.- From N. J. A. Sloane, Oct 08 2012
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry. J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
N. Elkies, G. Kuperberg, M. Larsen and J. Propp, Alternating sign matrices and domino tilings, Journal of Algebraic Combinatorics {\bf 1}, 111-132, 219-234 (1992).
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.
D. Grensing, I. Carlsen and H.-Chr. Zapp, Some exact results for the dimer problem on plane lattices with non-standard boundaries, Phil. Mag. A {\bf 41} (1980), 777-781.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
F. Harary, Graph Theory. Addison-Wesley, Reading, MA, 1969, p. 178.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 3, Eq. (1.1.2).
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011; http://repository.wit.ie/1693/1/AoifeThesis.pdf
W. Jockusch, Perfect matchings and perfect squares. J. Combin. Theory Ser. A 67 (1994), no. 1, 100-115.
W. H. Mills, D. P. Robbins and H. Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A {\bf 34} (1983), 340-359.
J. Propp, Enumeration of matchings: problems and progress, in: New perspectives in geometric combinatorics, L. Billera et al., eds., Mathematical Sciences Research Institute series, vol. 38, Cambridge University Press, 1999.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..50
F. Ardila and R. P. Stanley, Tilings
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
M. Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103.
M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97
S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem
H. Helfgott and I. M. Gessel, Enumeration of tilings of diamonds and hexagons with defects
E. H. Kuo, Applications of graphical condensation for enumerating matchings and tilings
G. Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
J. Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
J. Propp and R. P. Stanley, Domino tilings with barriers
S. S. Skiena, Generating graphs
D. E. Speyer, Perfect matchings and the octahedral recurrence
R. P. Stanley, A combinatorial miscellany
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Labeled Graph
Eric Weisstein's World of Mathematics, Symmetric Matrix
Index entries for sequences related to dominoes
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FORMULA
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Sequence is given by the Hankel transform of A001003 (Schroeder's numbers)= 1, 1, 3, 11, 45, 197, 903, ...; example : det([1, 1, 3, 11; 1, 3, 11, 45; 3, 11, 45, 197; 11, 45, 197, 903]) = 2^6 = 64 . - Philippe Deléham, Mar 02 2004
a(n)=2^floor(n^2/2)/2^floor(n/2). - Paul Barry, Oct 04 2004
G.f. satisfies: A(x) = 1 + x*A(2x). [From Paul D. Hanna, Dec 04 2009]
a(n) = 2 * a(n-1)^2 / a(n-2). - Michael Somos, Dec 30 2012
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MATHEMATICA
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Join[{1}, 2^Accumulate[Range[0, 20]]] (* Harvey P. Dale, May 09 2013 *)
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PROG
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(PARI) a(n)=1<<binomial(n, 2) \\ Charles R Greathouse IV, Jun 15 2011
(Maxima) A006125(n):=2^(n*(n-1)/2)$ makelist(A006125(n), n, 0, 30); /* Martin Ettl, Oct 24 2012 */
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CROSSREFS
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Cf. A000568 for the unlabeled analogue, A006129, A053763, A006253, A004003.
Cf. A001187 (connected labeled graphs).
Cf. A095340, A103904, A005329, A114604.
A158474 [From Gary W. Adamson, Mar 20 2009]
Cf. A136652 (log). [From Paul D. Hanna, Dec 04 2009]
Sequence in context: A139683 A139684 A139685 * A193753 A006119 A073113
Adjacent sequences: A006122 A006123 A006124 * A006126 A006127 A006128
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KEYWORD
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nonn,easy,nice,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Vladeta Jovovic, Apr 09 2000
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STATUS
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approved
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