A362171 -id:A362171 - cp's OEIS frontend

A362167 a(n) = the hypergraph Catalan number C_2(n).

1, 1, 6, 57, 678, 9270, 139968, 2285073, 39871926, 739129374, 14521778820, 302421450474, 6687874784484, 157491909678168, 3961138908376692, 106663881061254465, 3078671632202791782, 95213375569840078422, 3149291101933230285924, 111073721303120881912686

Offset: 0

Peter Bala, Apr 10 2023

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. When m = 1 we get the sequence of Catalan numbers A000108. The present sequence is the case m = 2.
Let T(n) be the set of unlabeled trees on n vertices (see A000055). Let T be a tree in T(n+1), and let v be a vertex of T. Then an a(m,T)-tour beginning at v is a walk that begins and ends at v and traverses each edge of T exactly 2*m times. We denote by a(m,T)(v) the number of a(m,T)-tours beginning at v.
The hypergraph Catalan numbers C_m(n) are defined by C_m(n) = Sum_{trees T in T(n+1)} Sum_{vertices v in T} a(m,T)(v)/|Aut(T)|, where Aut(T) denotes the automorphism group of the tree T.
Gunnells gives several combinatorial interpretations of the hypergraph Catalan numbers, a method to compute their generating functions to arbitrary precision and some conjectural asymptotics.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Column k=2 of A369288.

Cf. A000055, A000108, A362168, A362169, A362170, A362171, A362172.

Vec(HypCatColGf(2, 20)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 06 2024

a(n) ~ e^(3/2) * 2^(n+1) * n!/sqrt(Pi*n) (conjectural).

a(11) onwards from Andrew Howroyd, Jan 31 2024

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

Peter Bala, Apr 10 2023

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.

Andrew Howroyd, Table of n, a(n) for n = 0..80
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Column k=7 of A369288.

Cf. A000055, A000108, A362167, A362168, A362169, A362170, A362171.

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

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

Peter Bala, Apr 10 2023

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.

Andrew Howroyd, Table of n, a(n) for n = 0..200
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Column k=3 of A369288.

Cf. A000055, A000108, A362167, A362169, A362170, A362171, A362172.

Vec(HypCatColGf(3,15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024

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

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

Peter Bala, Apr 10 2023

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.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Column k=4 of A369288.

Cf. A000055, A000108, A362167, A362168, A362170, A362171, A362172.

Vec(HypCatColGf(4,15)) \\ HypCatColGf defined in A369288. - Andrew Howroyd, Feb 01 2024

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

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

Peter Bala, Apr 10 2023

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.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

Column k=5 of A369288.

Cf. A000055, A000108, A362167, A362168, A362169, A362171, A362172.

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

a(7) 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

Andrew Howroyd, Feb 01 2024

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.

			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 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021.

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

\\ 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,])) }

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.

cp's OEIS Frontend

A362167 a(n) = the hypergraph Catalan number C_2(n).

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

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

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

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

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

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