OFFSET
0,6
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.
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.
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.
EXAMPLE
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 ...
...
PROG
(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, ])) }
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 01 2024
STATUS
approved