OFFSET
0,3
COMMENTS
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
From James Propp: (Start)
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
|..| or |.....|
|..| o-----o
|..|
o--o
(End)
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, 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
The number of symmetric binary relations on an (n-1)-element set. - Peter Kagey, Feb 13 2021
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, Jun 12 2007
Equals right border of triangle A158474 (unsigned). - 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
a(n+1) is the number of symmetric binary matrices of size n X n. - Nathan J. Russell, Aug 30 2014
Let T_n be the n X n matrix with T_n(i,j) = binomial(2i + j - 3, j-1); then det(T_n) = a(n). - Tony Foster III, Aug 30 2018
k^(n*(n-1)/2) is the determinant of n X n matrix T_(i,j) = binomial(k*i + j - 3, j-1), in this case k=2. - Tony Foster III, May 12 2019
Let B_n be the n+1 X n+1 matrix with B_n(i, j) = Sum_{m=max(0, j-i)..min(j, n-i)} (binomial(i, j-m) * binomial(n-i, m) * (-1)^m), 0<=i,j<=n. Then det B_n = a(n+1). Also, deleting the first row and any column from B_n results in a matrix with determinant a(n). The matrices B_n have the following property: B_n * [x^n, x^(n-1) * y, x^(n-2) * y^2, ..., y^n]^T = [(x-y)^n, (x-y)^(n-1) * (x+y), (x-y)^(n-2) * (x+y)^2, ..., (x+y)^n]^T. - Nicolas Nagel, Jul 02 2019
a(n) is the number of positive definite (-1,1)-matrices of size n X n. - Eric W. Weisstein, Jan 03 2021
a(n) is the number of binary relations on a labeled n-set that are both total and antisymmetric. - José E. Solsona, Feb 05 2023
From Peter Bala, Jul 24 2026: (Start)
This sequence is an example of a polynomial recursive sequence as defined by Cadilhac et al: a(n+1) = P(a(n), b(n)) and b(n+1) = Q(a(n), b(n)) with a(0) = b(0) = 1, where P(x, y) = x*y, Q(x, y) = 2*y and b(n) = 2^n.
For prime p, the sequence obtained by reducing a(n) modulo p is ultimately periodic. For example, reducing the sequence modulo 11 gives the sequence [1, 1, 2, 8, 9, 1, 10, 2, 3, 9, 10, 10, 9, 3, 2, 10, 1, 9, 8, 2, 1, 1, 2, 8, 9, 1, 10, 2, 3, 9, 10, 10, 9, 3, 2, 10, 1, 9, 8, 2, 1, 1, 2, 8, 9, 1, 10, ...] with period 20 (= 2*(11 - 1)) starting at n = 1. (End)
REFERENCES
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 547 (Fig. 9.7), 573.
Miklos Bona, Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2025, pp. 292-293.
G. Everest, A. van der Poorten, I. Shparlinski, and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; p. 178.
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).
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).
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..50
Federico Ardila and Richard P. Stanley, Tilings, arXiv:math/0501170 [math.CO], 2005.
Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, J. Int. Seq. 15 (2012), Art. 12.4.8. - From N. J. A. Sloane, Oct 08 2012
Jose Bastidas, Félix Gélinas, Vincent Pilaud, Germain Poullot, Andrew Sack, and Eleni Tzanaki, Interval hypergraphic polytopes (or deformed associahedra), Tamari interval posets, and weeping willows, arXiv:2606.18376 [math.CO], 2026. See p. 16 (Table 2).
Anders Björner and Richard P. Stanley, A combinatorial miscellany, 2010.
Tobias Boege and Thomas Kahle, Construction Methods for Gaussoids, arXiv:1902.11260 [math.CO], 2019.
Taylor Brysiewicz and Fulvio Gesmundo, The Degree of Stiefel Manifolds, arXiv:1909.10085 [math.AG], 2019.
Michaël Cadilhac, Filip Mazowiecki, Charles Paperman, Michał Pilipczuk, and Géraud Sénizergues, On polynomial recursive sequences, arXiv:2002.08630 [cs.FL], 2020.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Int. Seq. 3 (2000), Art. 00.1.5.
Mihai Ciucu, Perfect matchings of cellular graphs, J. Algeb. Combin. 5 (1996), 87-103. See also Springer link.
Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77(1) (1997), 67-97.
Thierry de la Rue and Élise Janvresse, Pavages aléatoires par touillage de dominos, Images des Mathématiques, CNRS, 2023. In French.
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part I, J. Algeb. Combin. 1(2) (1992), 111-132.
Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating sign matrices and domino tilings. Part II, J. Algeb. Combin. 1(3) (1992), 219-234.
Sen-Peng Eu and Tung-Shan Fu, A simple proof of the Aztec diamond theorem, arXiv:math/0412041 [math.CO], 2004.
John Greene and Clayton Higgins, Maximal Subset Sums in a Group of Order a Power of 2, J. Int. Seq. 28 (2025), Art. 25.7.8. See pp. 3, 14.
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, 41(5) (1980), 777-781.
Harald Helfgott and Ira M. Gessel, Enumeration of tilings of diamonds and hexagons with defects, arXiv:math/9810143 [math.CO], 1998.
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.
Pakawut Jiradilok, Some Combinatorial Formulas Related to Diagonal Ramsey Numbers, arXiv:2404.02714 [math.CO], 2024. See p. 19.
William Jockusch, Perfect matchings and perfect squares, J. Combin. Theory Ser. A 67(1) (1994), 100-115.
Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoret. Comp. Sci. 319(1-3) (2004), 29-57. See also arXiv preprint, arXiv:math/0304090 [math.CO], 2003.
C. L. Mallows & N. J. A. Sloane, Emails, May 1991.
W. H. Mills, David P. Robbins, and Howard Rumsey, Jr., Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (1983), 340-359.
Götz Pfeiffer, Counting Transitive Relations, J. Int. Seq. 7 (2004), Art. 04.3.2.
James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics.
James Propp, Enumeration of matchings: problems and progress, arXiv:math/9904150 [math.CO], 1999.
James Propp, Lessons I learned from Richard Stanley, arXiv preprint [math.CO], 2015.
James Propp and R. P. Stanley, Domino tilings with barriers, arXiv:math/9801067 [math.CO], 1998.
Steven S. Skiena, Generating graphs.
David E. Speyer, Perfect matchings and the octahedron recurrence, J. Algeb. Combin. 25(3) (2007), 309-348. See also arXiv preprint, arXiv:math/0402452 [math.CO], 2004.
Jan Tóth and Ondřej Kuželka, Complexity of Weighted First-Order Model Counting in the Two-Variable Fragment with Counting Quantifiers: A Bound to Beat, arXiv:2404.12905 [cs.LO], 2024. See p. 24.
Herman Tulleken, Polyominoes 2.2: How they fit together, (2019).
Eric Weisstein's World of Mathematics, Connected Graph.
Eric Weisstein's World of Mathematics, Labeled Graph.
Eric Weisstein's World of Mathematics, Symmetric Matrix.
Doron Zeilberger, Dave Robbins's Art of Guessing, Adv. in Appl. Math. 34 (2005), 939-954.
FORMULA
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). - Paul D. Hanna, Dec 04 2009
a(n) = 2 * a(n-1)^2 / a(n-2). - Michael Somos, Dec 30 2012
G.f.: G(0)/x - 1/x, where G(k) = 1 + 2^(k-1)*x/(1 - 1/(1 + 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 26 2013
E.g.f. satisfies A'(x) = A(2x). - Geoffrey Critzer, Sep 07 2013
Sum_{n>=1} 1/a(n) = A299998. - Amiram Eldar, Oct 27 2020
a(n) = s_lambda(1,1,...,1) where s is the Schur polynomial in n variables and lambda is the partition (n,n-1,n-2,...,1). - Leonid Bedratyuk, Feb 06 2022
a(n) = Product_{1 <= j <= i <= n-1} (i + j)/(2*i - 2*j + 1). Cf. A007685. - Peter Bala, Oct 25 2024
EXAMPLE
From Gus Wiseman, Feb 11 2021: (Start)
This sequence counts labeled graphs on n vertices. For example, the a(0) = 1 through a(2) = 8 graph edge sets are:
{} {} {} {}
{12} {12}
{13}
{23}
{12,13}
{12,23}
{13,23}
{12,13,23}
This sequence also counts labeled graphs with loops on n - 1 vertices. For example, the a(1) = 1 through a(3) = 8 edge sets are the following. A loop is represented as an edge with two equal vertices.
{} {} {}
{11} {11}
{12}
{22}
{11,12}
{11,22}
{12,22}
{11,12,22}
(End)
MATHEMATICA
Join[{1}, 2^Accumulate[Range[0, 20]]] (* Harvey P. Dale, May 09 2013 *)
(* Alternative: *)
Table[2^(n (n - 1) / 2), {n, 0, 20}] (* Vincenzo Librandi, Jul 03 2019 *)
(* Alternative: *)
Table[2^Binomial[n, 2], {n, 0, 15}] (* Eric W. Weisstein, Jan 03 2021 *)
(* Alternative: *)
2^Binomial[Range[0, 15], 2] (* Eric W. Weisstein, Jan 03 2021 *)
(* Alternative: *)
Prepend[Table[Count[Tuples[{0, 1}, {n, n}], _?SymmetricMatrixQ], {n, 5}], 1] (* Eric W. Weisstein, Jan 03 2021 *)
(* Alternative: *)
Prepend[Table[Count[Tuples[{-1, 1}, {n, n}], _?PositiveDefiniteMatrixQ], 1], {n, 4}] (* Eric W. Weisstein, Jan 03 2021 *)
PROG
(PARI) a(n)=1<<binomial(n, 2) \\ Charles R Greathouse IV, Jun 15 2011
(Magma) [2^(n*(n-1) div 2): n in [0..20]]; // Vincenzo Librandi, Jul 03 2019
(Haskell) [2^(n*(n-1) `div` 2) | n <- [0..20]] -- José E. Solsona, Feb 05 2023
(Python)
def A006125(n): return 1<<(n*(n-1)>>1) # Chai Wah Wu, Nov 09 2023
CROSSREFS
Cf. A001187 (connected labeled graphs).
Cf. A158474. - Gary W. Adamson, Mar 20 2009
Cf. A136652 (log). - Paul D. Hanna, Dec 04 2009
The directed version is A002416.
The covering case is A006129.
Row sums of A143543.
The case of connected edge set is A287689.
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Apr 09 2000
STATUS
approved
