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A210586
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Triangle T(n,k) read by rows: T(n,k) is the number of rooted hypertrees on n labeled vertices with k hyperedges, n >= 2, k >= 1.
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5
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2, 3, 9, 4, 48, 64, 5, 175, 750, 625, 6, 540, 5400, 12960, 7776, 7, 1519, 30870, 156065, 252105, 117649, 8, 4032, 154112, 1433600, 4587520, 5505024, 2097152, 9, 10287, 704214, 11160261, 62001450, 141363306, 133923132, 43046721, 10, 25500, 3025000, 77700000, 695100000, 2646000000, 4620000000, 3600000000, 1000000000
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OFFSET
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2,1
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COMMENTS
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A hypergraph H is a pair (V,E) consisting of a finite set V of vertices and a set E of hyperedges given by subsets of V containing at least two elements. A walk in a hypergraph H connecting vertices v0 and vn is a sequence v0, e1, v1, e2, ... , v(n-1), en, vn, where each vi is in V and each ei is in E and for each ei the set {v(i-1),vi} is contained in ei. If for every pair of vertices v and v0 there is a walk in H starting at v and ending at v0 then H is called connected. A walk is a cycle if it contains at least two edges, all of the ei are distinct and all of the vi are distinct except v0 = vn. A connected hypergraph with no cycles is called a hypertree. A rooted hypertree is a hypertree in which one particular vertex is selected as being the root. For the enumeration of unrooted hypertrees see A210587.
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LINKS
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FORMULA
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T(n,k) = n^k*Stirling2(n-1,k). T(n,k) = n*A210587(n,k).
E.g.f. A(x,t) = t + 2*x*t^2/2! + (3*x + 9*x^2)*t^3/3! + ... satisfies A(x,t) = t*exp(x*(exp(A(x,t)) - 1)).
Dobinski-type formula for the row polynomials: R(n,x) = exp(-n*x)*sum {k = 0..inf} n^k*k^(n-1)x^k/k!.
The e.g.f. is essentially the series reversion of t/F(x,t) w.r.t. t, where F(x,t) = exp(x*(exp(t) - 1)) is the e.g.f. of the Stirling numbers of the second kind A048993. - Peter Bala, Oct 28 2015
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EXAMPLE
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Triangle begins
.n\k.|....1.....2......3.......4.......5.......6
= = = = = = = = = = = = = = = = = = = = = = = = =
..2..|....2
..3..|....3.....9
..4..|....4....48.....64
..5..|....5...175....750.....625
..6..|....6...540...5400...12960....7776
..7..|....7..1519..30870..156065..252105..117649
...
Example of a hypertree with two hyperedges, one a 2-edge {3,4) and one a 3-edge{1,2,3}.
........__________........................
......./..........\.______................
......|....1...../.\......\...............
......|.........|.3.|....4.|..............
......|....2.....\./______/...............
.......\__________/.......................
..........................................
T(4,2) = 48. The twelve unrooted hypertrees on 4 vertices {1,2,3,4} with 2 hyperedges (one a 2-edge and one a 3-edge) have hyperedges:
{1,2,3} and {3,4); {1,2,3} and {2,4); {1,2,3} and {1,4);
{1,2,4} and {1,3); {1,2,4} and {2,3); {1,2,4} and {3,4);
{1,3,4} and {1,2); {1,3,4} and {2,3); {1,3,4} and {2,4);
{2,3,4} and {1,2); {2,3,4} and {1,3); {2,3,4} and {1,4).
Choosing one of the four vertices as the root gives a total of 4x12 = 48 rooted hypertrees on 4 vertices.
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MAPLE
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with(combinat):
A210586 := (n, k) -> n^k*stirling2(n-1, k):
for n from 2 to 10 do seq(A210586(n, k), k = 1..n-1) end do;
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PROG
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(PARI) T(n, k) = {n^k*stirling(n-1, k, 2)}
for(n=2, 10, for(k=1, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Aug 28 2018
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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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