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A072590 Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals. 6
1, 1, 1, 1, 4, 1, 1, 12, 12, 1, 1, 32, 81, 32, 1, 1, 80, 432, 432, 80, 1, 1, 192, 2025, 4096, 2025, 192, 1, 1, 448, 8748, 32000, 32000, 8748, 448, 1, 1, 1024, 35721, 221184, 390625, 221184, 35721, 1024, 1, 1, 2304, 139968, 1404928, 4050000, 4050000 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

REFERENCES

J. W. Moon, Counting Labeled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.66.

LINKS

T. D. Noe, Antidiagonals d=1..50, flattened

H. I. Scoins, The number of trees with nodes of alternate parity, Proc. Cambridge Philos. Soc. 58 (1962) 12-16.

FORMULA

T(n, k) = n^(k-1) * k^(n-1).

E.g.f. A(x,y) - 1, where: A(x,y) = exp( x*exp( y*A(x,y) ) ) = Sum_{n>=0} Sum_{k=0..n} (n-k)^k * (k+1)^(n-k-1) * x^(n-k)/(n-k)! * y^k/k!. - Paul D. Hanna, Jan 22 2019

EXAMPLE

1; 1,1; 1,4,1; 1,12,12,1; 1,32,81,32,1; 1,80,432,432,80,1; ...

MATHEMATICA

t[n_, k_] := n^(k-1) * k^(n-1); Table[ t[n-k+1, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 21 2013 *)

PROG

(PARI) {T(n, k) = if( n<1 || k<1, 0, n^(k-1) * k^(n-1))}

CROSSREFS

A068087(n)=T(n, n). Cf. A001787, A069996.

Sequence in context: A213166 A168619 A099759 * A111636 A220688 A146990

Adjacent sequences:  A072587 A072588 A072589 * A072591 A072592 A072593

KEYWORD

nonn,tabl,easy,nice

AUTHOR

Michael Somos, Jun 23 2002

EXTENSIONS

Scoins reference from Philippe Deléham, Dec 22 2003

STATUS

approved

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Last modified March 18 12:10 EDT 2019. Contains 321283 sequences. (Running on oeis4.)