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%I #13 Sep 26 2019 04:19:32
%S 1,3,1,16,1,125,15,1,1296,1,16807,735,140,1,262144,1,4782969,76545,
%T 1890,1,100000000,112000,1,2357947691,13835745,33264,1,61917364224,1,
%U 1792160394037,3859590735,270670400,35135100,720720,1,56693912375296,1,1946195068359375
%N Irregular triangle read by rows where T(n,d) for d|n is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices.
%C A hypertree is a connected hypergraph of density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices. A hypergraph is k-uniform if its edges all have size k. The span of a hypertree is the union of its edges.
%H Alois P. Heinz, <a href="/A326374/b326374.txt">Rows n = 1..185, flattened</a>
%F T(n, d) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).
%e Triangle begins:
%e 1
%e 3 1
%e 16 1
%e 125 15 1
%e 1296 1
%e 16807 735 140 1
%e 262144 1
%e 4782969 76545 1890 1
%e 100000000 112000 1
%e 2357947691 13835745 33264 1
%e The a(4,2) = 15 hypertrees:
%e {{1,4,5},{2,3,5}}
%e {{1,4,5},{2,3,4}}
%e {{1,3,5},{2,4,5}}
%e {{1,3,5},{2,3,4}}
%e {{1,3,4},{2,4,5}}
%e {{1,3,4},{2,3,5}}
%e {{1,2,5},{3,4,5}}
%e {{1,2,5},{2,3,4}}
%e {{1,2,5},{1,3,4}}
%e {{1,2,4},{3,4,5}}
%e {{1,2,4},{2,3,5}}
%e {{1,2,4},{1,3,5}}
%e {{1,2,3},{3,4,5}}
%e {{1,2,3},{2,4,5}}
%e {{1,2,3},{1,4,5}}
%p T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
%p seq(T(n), n=1..20); # _Alois P. Heinz_, Aug 21 2019
%t Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]
%Y Row sums are A320444.
%Y Column d = 1 is A000272.
%Y Cf. A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.
%K nonn,look,tabf
%O 1,2
%A _Gus Wiseman_, Jul 03 2019
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