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A369288
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Array read by antidiagonals: A(n,k) = the hypergraph Catalan number C_k(n), n >= 0, k >= 1.
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6
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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
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OFFSET
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0,6
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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 ...
...
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PROG
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(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, ])) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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