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 A072590 Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals. 10
 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 Taylor Brysiewicz and Aida Maraj, Lawrence Lifts, Matroids, and Maximum Likelihood Degrees, arXiv:2310.13064 [math.CO], 2023. See p. 13. H. I. Scoins, The number of trees with nodes of alternate parity, Proc. Cambridge Philos. Soc. 58 (1962) 12-16. Eric Weisstein's World of Mathematics, Complete Bipartite Graph Eric Weisstein's World of Mathematics, Spanning Tree 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 From Andrew Howroyd, Oct 29 2019: (Start) Array begins: ============================================================ n\k | 1 2 3 4 5 6 7 ----+------------------------------------------------------- 1 | 1 1 1 1 1 1 1 ... 2 | 1 4 12 32 80 192 448 ... 3 | 1 12 81 432 2025 8748 35721 ... 4 | 1 32 432 4096 32000 221184 1404928 ... 5 | 1 80 2025 32000 390625 4050000 37515625 ... 6 | 1 192 8748 221184 4050000 60466176 784147392 ... 7 | 1 448 35721 1404928 37515625 784147392 13841287201 ... ... (End) 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 Columns 2..3 are A001787, A069996. Main diagonal is A068087. Antidiagonal sums are A132609. Cf. A070285, A328887, A328888. Sequence in context: A168619 A099759 A350819 * A350745 A111636 A220688 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 June 20 07:20 EDT 2024. Contains 373512 sequences. (Running on oeis4.)