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A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).

%I A326374 #16 Mar 05 2025 22:04:36
%S A326374 1,3,1,16,1,125,15,1,1296,1,16807,735,140,1,262144,1,4782969,76545,
%T A326374 1890,1,100000000,112000,1,2357947691,13835745,33264,1,61917364224,1,
%U A326374 1792160394037,3859590735,270670400,35135100,720720,1,56693912375296,1,1946195068359375
%N A326374 Irregular triangle read by rows where T(n,k) is the number of (d + 1)-uniform hypertrees spanning n + 1 vertices, where d = A027750(n,k).
%C A326374 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 A326374 Alois P. Heinz, <a href="/A326374/b326374.txt">Rows n = 1..185, flattened</a>
%F A326374 T(n, k) = n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1), where d = A027750(n, k).
%e A326374 Triangle begins:
%e A326374            1
%e A326374            3          1
%e A326374           16          1
%e A326374          125         15          1
%e A326374         1296          1
%e A326374        16807        735        140          1
%e A326374       262144          1
%e A326374      4782969      76545       1890          1
%e A326374    100000000     112000          1
%e A326374   2357947691   13835745      33264          1
%e A326374 The T(4,2) = 15 hypertrees:
%e A326374   {{1,4,5},{2,3,5}}
%e A326374   {{1,4,5},{2,3,4}}
%e A326374   {{1,3,5},{2,4,5}}
%e A326374   {{1,3,5},{2,3,4}}
%e A326374   {{1,3,4},{2,4,5}}
%e A326374   {{1,3,4},{2,3,5}}
%e A326374   {{1,2,5},{3,4,5}}
%e A326374   {{1,2,5},{2,3,4}}
%e A326374   {{1,2,5},{1,3,4}}
%e A326374   {{1,2,4},{3,4,5}}
%e A326374   {{1,2,4},{2,3,5}}
%e A326374   {{1,2,4},{1,3,5}}
%e A326374   {{1,2,3},{3,4,5}}
%e A326374   {{1,2,3},{2,4,5}}
%e A326374   {{1,2,3},{1,4,5}}
%p A326374 T:= n-> seq(n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1), d=numtheory[divisors](n)):
%p A326374 seq(T(n), n=1..20);  # _Alois P. Heinz_, Aug 21 2019
%t A326374 Table[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{n,10},{d,Divisors[n]}]
%Y A326374 Row sums are A320444.
%Y A326374 Column d = 1 is A000272.
%Y A326374 Cf. A030019, A027750, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.
%K A326374 nonn,look,tabf
%O A326374 1,2
%A A326374 _Gus Wiseman_, Jul 03 2019
%E A326374 Edited by _Peter Munn_, Mar 05 2025