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

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

Author

Andrew Howroyd, Feb 01 2024

Keywords

• nonn
• tabl

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

• 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.

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.