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%I #42 Oct 30 2023 14:36:33
%S 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,
%T 2025,192,1,1,448,8748,32000,32000,8748,448,1,1,1024,35721,221184,
%U 390625,221184,35721,1024,1,1,2304,139968,1404928,4050000,4050000
%N Table T(n,k) giving number of spanning trees in complete bipartite graph K(n,k), read by antidiagonals.
%D J. W. Moon, Counting Labeled Trees, Issue 1 of Canadian mathematical monographs, Canadian Mathematical Congress, 1970.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 5.66.
%H T. D. Noe, <a href="/A072590/b072590.txt">Antidiagonals d=1..50, flattened</a>
%H Taylor Brysiewicz and Aida Maraj, <a href="https://arxiv.org/abs/2310.13064">Lawrence Lifts, Matroids, and Maximum Likelihood Degrees</a>, arXiv:2310.13064 [math.CO], 2023. See p. 13.
%H H. I. Scoins, <a href="https://doi.org/10.1017/S0305004100036173">The number of trees with nodes of alternate parity</a>, Proc. Cambridge Philos. Soc. 58 (1962) 12-16.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SpanningTree.html">Spanning Tree</a>
%F T(n, k) = n^(k-1) * k^(n-1).
%F 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
%e From _Andrew Howroyd_, Oct 29 2019: (Start)
%e Array begins:
%e ============================================================
%e n\k | 1 2 3 4 5 6 7
%e ----+-------------------------------------------------------
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 1 4 12 32 80 192 448 ...
%e 3 | 1 12 81 432 2025 8748 35721 ...
%e 4 | 1 32 432 4096 32000 221184 1404928 ...
%e 5 | 1 80 2025 32000 390625 4050000 37515625 ...
%e 6 | 1 192 8748 221184 4050000 60466176 784147392 ...
%e 7 | 1 448 35721 1404928 37515625 784147392 13841287201 ...
%e ...
%e (End)
%t 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 *)
%o (PARI) {T(n, k) = if( n<1 || k<1, 0, n^(k-1) * k^(n-1))}
%Y Columns 2..3 are A001787, A069996.
%Y Main diagonal is A068087.
%Y Antidiagonal sums are A132609.
%Y Cf. A070285, A328887, A328888.
%K nonn,tabl,easy,nice
%O 1,5
%A _Michael Somos_, Jun 23 2002
%E Scoins reference from _Philippe Deléham_, Dec 22 2003
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