cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A006125 a(n) = 2^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 2, 8, 64, 1024, 32768, 2097152, 268435456, 68719476736, 35184372088832, 36028797018963968, 73786976294838206464, 302231454903657293676544, 2475880078570760549798248448, 40564819207303340847894502572032, 1329227995784915872903807060280344576
Offset: 0

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Author

N. J. A. Sloane

Keywords

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

Examples

			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)

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 547 (Fig. 9.7), 573.
  • 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

Crossrefs

Cf. A000568 for the unlabeled analog, A053763, A006253, A004003.
Cf. A001187 (connected labeled graphs).
Cf. A095340, A103904, A005329, A114604, A299998.
Cf. A158474. - Gary W. Adamson, Mar 20 2009
Cf. A136652 (log). - Paul D. Hanna, Dec 04 2009
The unlabeled version is A000088, or A002494 without isolated vertices.
The directed version is A002416.
The covering case is A006129.
The version for hypergraphs is A058891, or A016031 without singletons.
Row sums of A143543.
The case of connected edge set is A287689.
Cf. A000169, A000295, A036442, A059167, A100743, A136284, A327078.

Programs

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

Extensions

More terms from Vladeta Jovovic, Apr 09 2000

A379706 Number of nonempty labeled antichains of subsets of [n] such that the largest subset is of size 2.

Original entry on oeis.org

0, 0, 1, 10, 97, 1418, 40005, 2350474, 286192257, 71213783154, 35883905262757, 36419649682704418, 74221659280476132145, 303193505953871645554778, 2480118046704094643352342117, 40601989176407026666590990389338, 1329877330167226219547875498464450945, 87134888326188320631048795061602782878050
Offset: 0

Views

Author

John Tyler Rascoe, Dec 30 2024

Keywords

Examples

			a(2) = 1: {{1,2}}.
a(3) = 10: {{1,2}}, {{1,3}}, {{2,3}}, {{1,2},{3}}, {{1,3},{2}}, {{2,3},{1}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.

Crossrefs

Cf. A000225, A000372, A036239, A126883, A379707, (column k=2 of A379712).

Programs

  • Python
    from math import comb
def A379706(n): return sum(comb(n,i)*(2**comb(n-i,2) - 1) for i in range(n-1))

Formula

a(n) = Sum_{i=0..n-2} binomial(n,i) * (2^binomial(n-i,2) - 1).

A059328 Table T(n,k) = T(n - 1,k) + T(n,k - 1) + T(n - 1,k)*T(n,k - 1) starting with T(0,0)=1, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 63, 15, 1, 1, 31, 1023, 1023, 31, 1, 1, 63, 32767, 1048575, 32767, 63, 1, 1, 127, 2097151, 34359738367, 34359738367, 2097151, 127, 1, 1, 255, 268435455, 72057594037927935, 1180591620717411303423, 72057594037927935, 268435455, 255, 1
Offset: 0

Views

Author

Henry Bottomley, Jan 26 2001

Keywords

Comments

In binary representation T(n,k) is the concatenation of T(n-1,k-1) and T(n-1,k), 0Reinhard Zumkeller, Jan 23 2003

Examples

			Triangle T(n,k) begins:
  1;
  1,  1;
  1,  3,     1;
  1,  7,     7,       1;
  1, 15,    63,      15,     1;
  1, 31,  1023,    1023,    31,  1;
  1, 63, 32767, 1048575, 32767, 63, 1;
  ...

Links

Crossrefs

Columns k=0-2 give: A000012, A000225, A126883(n-1).
Row sums give A306020.
Cf. A007318.

Programs

  • Mathematica
    Table[2^(Binomial[n, k]) - 1, {n, 0, 5}, {k, 0, n}] (* G. C. Greubel, Jan 07 2017 *)
  • Python
    from math import comb, isqrt
def A059328(n): return (1<Chai Wah Wu, Apr 30 2025

Formula

T(n, k) = 2^C(n+k, n)-1; a(n) = 2^A007318(n)-1.
If U(n, k) := 1 + T(n, k), then U(n, k) = U(n-1, k) * U(n, k-1). - Michael Somos, Jan 07 2017

A371033 a(n) is the integer whose binary expansion starts with 1 and such that the runs of identical bits have lengths n, n-1, n-2, ..., 3, 2, 1.

Original entry on oeis.org

1, 6, 57, 966, 31801, 2065350, 266370105, 68453106630, 35115918982201, 35993681099981766, 73750982613738224697, 302157703921043555451846, 2475577920866839506242796601, 40562343629382474008388259775430, 1329187433441286490429798672020569145
Offset: 1

Views

Author

Clark Kimberling, Mar 18 2024

Keywords

Examples

			Representations as binary words (as in A371032) have decreasing runlengths:
    1:  1
    6:  110
   57:  111001
  966:  1111000110  (runlengths 4,3,2,1)

Crossrefs

Cf. A006125, A007088, A065760, A126883, A371032 (binary version).

Programs

  • Maple
    a:= n-> Bits[Join]([seq((1-(n-i) mod 2)$i, i=1..n)]):
seq(a(n), n=1..15);  # Alois P. Heinz, Jul 09 2024
  • Mathematica
    Map[FromDigits[#, 2] &, Table[Flatten[Map[ConstantArray[Mod[#, 2], n + 1 - #] &, Range[n]]], {n, 16}]]    (* Peter J. C. Moses, Mar 08 2024 *)
  • Python
    def A371033(n):
    c = 0
    for i in range(n):
        c <<= n-i
        if i&1^1:
            c += (1<Chai Wah Wu, Mar 18 2024

Formula

a(n) == n (mod 2). - Alois P. Heinz, Jul 09 2024
a(n) = 2^(n*(n+1)/2) - 1 - a(n-1). - Robert Israel, Jul 09 2024

Extensions

New name from Michel Marcus, Jul 09 2024
a(15) corrected by Alois P. Heinz, Jul 09 2024

A206601 3^(n(n+1)/2) - 1.

Original entry on oeis.org

0, 2, 26, 728, 59048, 14348906, 10460353202, 22876792454960, 150094635296999120, 2954312706550833698642, 174449211009120179071170506, 30903154382632612361920641803528, 16423203268260658146231467800709255288, 26183890704263137277674192438430182020124346
Offset: 0

Views

Author

Ivan N. Ianakiev, Feb 10 2012

Keywords

Comments

There are n cities located on the vertices of a convex n-gon and 2 types of communication lines available. Any city can be connected to any other by only one communication line (that can be of any type). A network exists if at least 2 cities are connected by a communication line. The sequence shows how many different networks a(n) can be built. In general, if the number of communication-line types is c, then a(n) = (c+1)^(n(n+1)/2)-1. Thus other sequences of this type can be generated.

Examples

			In the case of 2 different types of communication lines and 4 cities, the number of different networks (connecting at least 2 cities) is 728.

Crossrefs

Cf. A000217, A126883.

Formula

a(n) = (3^A000217) - 1.
a(n) = A047656(n+1) - 1. - Omar E. Pol, Feb 18 2012
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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