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 1-10 of 10 results.

A362172 a(n) = the hypergraph Catalan number C_7(n).

Original entry on oeis.org

1, 1, 3432, 141858288, 40309820014464, 53321581727982247680, 238681094467043912358445056, 2924960829706245011243295851200512, 84750120431280677998861681616641721991168, 5208807724759446156144077076658272647436908396544
Offset: 0

Views

Author

Peter Bala, Apr 10 2023

Keywords

Comments

Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 7.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

Links

Crossrefs

Column k=7 of A369288.
Cf. A000055, A000108, A362167, A362168, A362169, A362170, A362171.

Formula

a(n) ~ sqrt(7)/4 * (7^6/6!)^n * n!^6/(Pi*n)^3 (conjectural).

Extensions

a(7) onwards from Andrew Howroyd, Feb 01 2024

A362168 a(n) = the hypergraph Catalan number C_3(n).

Original entry on oeis.org

1, 1, 20, 860, 57200, 5344800, 682612800, 118180104000, 27396820448000, 8312583863720000, 3209035788149600000, 1534218535286625760000, 888028389273314675200000, 611029957551257895664000000, 492466785518772137553984000000, 459270692175324078697443840000000
Offset: 0

Views

Author

Peter Bala, Apr 10 2023

Keywords

Comments

Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 3.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

Links

Crossrefs

Column k=3 of A369288.
Cf. A000055, A000108, A362167, A362169, A362170, A362171, A362172.

Programs

Formula

a(n) ~ sqrt(3) * (9/2)^n * n!^2/(Pi*n) (conjectural).

Extensions

a(9) onwards from Andrew Howroyd, Feb 01 2024

A362169 a(n) = the hypergraph Catalan number C_4(n).

Original entry on oeis.org

1, 1, 70, 15225, 7043750, 6327749750, 10411817136000, 29034031694460625, 126890003304310093750, 816448082514611102718750, 7379204202189710013311562500, 90369225128606332243844280406250, 1457163640851863433667228849319062500, 30217741884769257764596041337071409375000
Offset: 0

Views

Author

Peter Bala, Apr 10 2023

Keywords

Comments

Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 4.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

Links

Crossrefs

Column k=4 of A369288.
Cf. A000055, A000108, A362167, A362168, A362170, A362171, A362172.

Programs

Formula

a(n) ~ sqrt(2) * (32/3)^n * n!^3/(Pi*n)^(3/2) (conjectural).

Extensions

a(8) onwards from Andrew Howroyd, Feb 01 2024

A362170 a(n) = the hypergraph Catalan number C_5(n).

Original entry on oeis.org

1, 1, 252, 299880, 1112865264, 11126161436292, 255654847841227632, 11676346013544951854304, 953196481551725431240711680, 128864126679853773803689954958112, 27235509875891350493949247236459319296, 8599544533810439129313490410035564948257536
Offset: 0

Views

Author

Peter Bala, Apr 10 2023

Keywords

Comments

Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 5.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

Links

Crossrefs

Column k=5 of A369288.
Cf. A000055, A000108, A362167, A362168, A362169, A362171, A362172.

Formula

a(n) ~ sqrt(5)/2 * (5^4/24)^n * n!^4/(Pi*n)^2 (conjectural).

Extensions

a(7) onwards from Andrew Howroyd, Feb 01 2024

A362171 a(n) = the hypergraph Catalan number C_6(n).

Original entry on oeis.org

1, 1, 924, 6358044, 203356067376, 23345633108619360, 7484535614458774428480, 5583028528736289502562408256, 8547031978688473343843434600852224, 24503310825110075324451531207978424853568, 122607946140627185219752569884701085604290069760
Offset: 0

Views

Author

Peter Bala, Apr 10 2023

Keywords

Comments

Let m >= 1. The sequence of hypergraph Catalan numbers {C_m(n): n >= 0} is defined in terms of counting walks on trees, weighted by the orders of their automorphism groups. See Gunnells. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 6.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

Links

Crossrefs

Column k=6 of A369288.
Cf. A000055, A000108, A362167, A362168, A362169, A362170, A362172.

Formula

a(n) ~ sqrt(3)/2 * (6^5/5!)^n * n!^5/(Pi*n)^(5/2) (conjectural)

Extensions

a(6) onwards from Andrew Howroyd, Feb 01 2024

A369288 Array read by antidiagonals: A(n,k) = the hypergraph Catalan number C_k(n), n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 6, 5, 1, 1, 20, 57, 14, 1, 1, 70, 860, 678, 42, 1, 1, 252, 15225, 57200, 9270, 132, 1, 1, 924, 299880, 7043750, 5344800, 139968, 429, 1, 1, 3432, 6358044, 1112865264, 6327749750, 682612800, 2285073, 1430, 1, 1, 12870, 141858288, 203356067376, 11126161436292, 10411817136000, 118180104000, 39871926, 4862
Offset: 0

Views

Author

Andrew Howroyd, Feb 01 2024

Keywords

Comments

Definition (from A362167): Let Trees(n) be the set of unlabeled trees on n vertices (see A000055). Let T be in Trees(n+1), and let v be a vertex of T. Then a (k,T)-tour beginning at v is a walk that begins and ends at v and traverses each edge of T exactly 2*k times. We denote by N(k,T)(v) the number of (k,T)-tours beginning at v.
The hypergraph Catalan numbers C_k(n) are defined by C_k(n) = Sum_{trees T in T(n+1)} Sum_{vertices v in T} N(k,T)(v)/|Aut(T)|, where Aut(T) denotes the automorphism group of the tree T.
See the Gunnells reference for a full definition and additional information.

Examples

			Array begins:
n/k|   1       2            3              4                  5 ...
---+-----------------------------------------------------------------
 0 |   1       1            1              1                  1 ...
 1 |   1       1            1              1                  1 ...
 2 |   2       6           20             70                252 ...
 3 |   5      57          860          15225             299880 ...
 4 |  14     678        57200        7043750         1112865264 ...
 5 |  42    9270      5344800     6327749750     11126161436292 ...
 6 | 132  139968    682612800 10411817136000 255654847841227632 ...
  ...

Links

Crossrefs

Columns k=1..7 are A000108, A362167, A362168, A362169, A362170, A362171, A362172.
Row 2 is A000984.
Cf. A000055, A060540.

Programs

  • PARI
    \\ here L(k,n) is k-th column of A060540 as g.f.
L(k,n)={sum(n=1, n, (n*k)!*x^n/(k!^n*n!), O(x*x^n))}
HypCatColGf(k,n)={my(p=L(k,n)); 1 + subst(p, x, serreverse(x^2/p))}
M(n,m=n+1)={Mat(vector(m, k, Col(HypCatColGf(k,n))))}
{ my(A=M(7,5)); for(i=1, matsize(A)[1], print(A[i,])) }

Formula

G.f. of column k: 1 + B_k(Series_Reversion(x^2/B_k(x))) where B_k(x) is the g.f. of column k of A060540.

A362214 a(n) = the hypergraph Fuss-Catalan number FC_(2,2)(n).

Original entry on oeis.org

1, 1, 144, 1341648, 693520980336
Offset: 0

Views

Author

Peter Bala, Apr 11 2023

Keywords

Comments

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (2,2).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Links

Crossrefs

Cf. A000108, A355262, A362167, A362215, A362216, A362217.

A362215 a(n) = the hypergraph Fuss-Catalan number FC_(2,3)(n).

Original entry on oeis.org

1, 1, 480, 200225, 18527520, 45589896150400
Offset: 0

Views

Author

Peter Bala, Apr 11 2023

Keywords

Comments

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (2,3).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Links

Crossrefs

Cf. A000108, A355262, A362167, A362214, A362216, A362217.

A362216 a(n) = the hypergraph Fuss-Catalan number FC_(3,2)(n).

Original entry on oeis.org

1, 1, 11532, 628958939250, 163980917165716725552156
Offset: 0

Views

Author

Peter Bala, Apr 11 2023

Keywords

Comments

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (3,2).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Links

Crossrefs

Cf. A000108, A355262, A362167, A362214, A362215, A362217.

A362217 a(n) = the hypergraph Fuss-Catalan number FC_(3,3)(n).

Original entry on oeis.org

1, 1, 38440, 8272793255000, 9396808005460764741084000
Offset: 0

Views

Author

Peter Bala, Apr 11 2023

Keywords

Comments

Chavan et al. associate to each pair (r,m) of positive integers the sequence of hypergraph Fuss-Catalan numbers {FC_(r,m)(n) : n >= 0}. This is the case (r,m) = (3,3).
When m = 1, the sequence {FC_(r,1)(n) : n >= 0} is equivalent to the sequence of Fuss-Catalan numbers { (1/(r*n+1))*binomial((r+1)*n,n) : n >= 0}. Note that r = 1 corresponds to the Catalan numbers A000108. See A355262 for a table of Fuss-Catalan numbers.
When r = 1, the sequence {FC_(1,m)(n) : n >= 0} is equivalent to the sequence of hypergraph Catalan numbers {C_m(n) : n >= 0}. See A362167 - A362172 for the cases m = 2 through 7.

Links

Crossrefs

Cf. A000108, A355262, A362167, A362214, A362215, A362216.
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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