|
%I #8 Jan 09 2020 19:28:23
%S 1,1,1,1,6,1,1,18,18,1,1,46,132,46,1,1,110,696,696,110,1,1,254,3150,
%T 6728,3150,254,1,1,574,13086,51760,51760,13086,574,1,1,1278,51492,
%U 348048,632970,348048,51492,1278,1,1,2814,195180,2143736,6466980,6466980,2143736,195180,2814,1
%N Array read by antidiagonals: T(n,m) is the number of acyclic edge covers of the complete bipartite graph K_{n,m}.
%C In other words, the number of spanning forests of the complete bipartite graph K_{n,m} without isolated vertices.
%H Andrew Howroyd, <a href="/A328888/b328888.txt">Table of n, a(n) for n = 1..1275</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%F T(n,m) = A072590(n, m) + Sum_{i=1..n-1} Sum_{j=1, m-1} binomial(n-1, i-1) * binomial(m, j) * A072590(i, j) * T(n-i, m-j).
%e Array begins:
%e =============================================================
%e n\m | 1 2 3 4 5 6 7
%e ----+--------------------------------------------------------
%e 1 | 1 1 1 1 1 1 1 ...
%e 2 | 1 6 18 46 110 254 574 ...
%e 3 | 1 18 132 696 3150 13086 51492 ...
%e 4 | 1 46 696 6728 51760 348048 2143736 ...
%e 5 | 1 110 3150 51760 632970 6466980 58620030 ...
%e 6 | 1 254 13086 348048 6466980 96208632 1231832364 ...
%e 7 | 1 574 51492 2143736 58620030 1231832364 21634786586 ...
%e ...
%o (PARI)
%o T(n, m=n)={my(M=matrix(n, m), N=matrix(n, m, n, m, n^(m-1) * m^(n-1))); for(n=1, n, for(m=1, m, M[n,m] = N[n,m] + sum(i=1, n-1, sum(j=1, m-1, binomial(n-1, i-1)*binomial(m, j)*N[i,j]*M[n-i, m-j])))); M}
%o { my(A=T(7)); for(i=1, #A, print(A[i,])) }
%Y Column 2 is A328890.
%Y Main diagonal is A328889.
%Y Cf. A072590, A328887.
%K nonn,tabl
%O 1,5
%A _Andrew Howroyd_, Oct 29 2019
|
