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.
%I A369288 #22 Oct 15 2024 00:01:58
%S A369288 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,
%T A369288 57200,9270,132,1,1,924,299880,7043750,5344800,139968,429,1,1,3432,
%U A369288 6358044,1112865264,6327749750,682612800,2285073,1430,1,1,12870,141858288,203356067376,11126161436292,10411817136000,118180104000,39871926,4862
%N A369288 Array read by antidiagonals: A(n,k) = the hypergraph Catalan number C_k(n), n >= 0, k >= 1.
%C A369288 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.
%C A369288 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.
%C A369288 See the Gunnells reference for a full definition and additional information.
%H A369288 Andrew Howroyd, <a href="/A369288/b369288.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)
%H A369288 Paul E. Gunnells, <a href="https://arxiv.org/abs/2102.05121">Generalized Catalan numbers from hypergraphs</a>, arXiv:2102.05121 [math.CO], 2021.
%F A369288 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.
%e A369288 Array begins:
%e A369288 n/k| 1 2 3 4 5 ...
%e A369288 ---+-----------------------------------------------------------------
%e A369288 0 | 1 1 1 1 1 ...
%e A369288 1 | 1 1 1 1 1 ...
%e A369288 2 | 2 6 20 70 252 ...
%e A369288 3 | 5 57 860 15225 299880 ...
%e A369288 4 | 14 678 57200 7043750 1112865264 ...
%e A369288 5 | 42 9270 5344800 6327749750 11126161436292 ...
%e A369288 6 | 132 139968 682612800 10411817136000 255654847841227632 ...
%e A369288 ...
%o A369288 (PARI) \\ here L(k,n) is k-th column of A060540 as g.f.
%o A369288 L(k,n)={sum(n=1, n, (n*k)!*x^n/(k!^n*n!), O(x*x^n))}
%o A369288 HypCatColGf(k,n)={my(p=L(k,n)); 1 + subst(p, x, serreverse(x^2/p))}
%o A369288 M(n,m=n+1)={Mat(vector(m, k, Col(HypCatColGf(k,n))))}
%o A369288 { my(A=M(7,5)); for(i=1, matsize(A)[1], print(A[i,])) }
%Y A369288 Columns k=1..7 are A000108, A362167, A362168, A362169, A362170, A362171, A362172.
%Y A369288 Row 2 is A000984.
%Y A369288 Cf. A000055, A060540.
%K A369288 nonn,tabl
%O A369288 0,6
%A A369288 _Andrew Howroyd_, Feb 01 2024